Explanation:
Okay, so I need to find the side length of an equilateral triangle inscribed in the parabola $ x^2 = 8y $, with one vertex at the vertex of the parabola. Let me start by recalling some properties of parabolas and equilateral triangles.

First, the given parabola is $ x^2 = 8y $. This is a standard upward-opening parabola with vertex at the origin (0,0). The general form of such a parabola is $ x^2 = 4py $, where p is the focal length. Comparing, we have 4p = 8, so p = 2. Therefore, the focus is at (0,2), but maybe that's not directly relevant here. However, knowing the vertex is at (0,0) is important because one of the triangle's vertices is there.

Now, the triangle is equilateral, so all sides are equal. Let me denote the three vertices of the triangle as A, B, and C. Given that one of them is the vertex of the parabola, let's say point A is (0,0). Then points B and C are somewhere on the parabola.

So, we need to find coordinates for points B and C on the parabola such that triangle ABC is equilateral. Since the parabola is symmetric about the y-axis, maybe there's some symmetry in the triangle as well. If the triangle is equilateral, maybe points B and C are symmetric with respect to the y-axis? Let me consider that possibility.

Let me assume that points B and C are symmetric across the y-axis. So if point B is (a, b), then point C would be (-a, b). Since both points lie on the parabola $ x^2 = 8y $, substituting, we get for point B: $ a^2 = 8b $, so b = $ a^2 / 8 $.

Therefore, coordinates of B and C are (a, $ a^2 / 8 $) and (-a, $ a^2 / 8 $).

Now, the triangle ABC has vertices at (0,0), (a, $ a^2 / 8 $), and (-a, $ a^2 / 8 $). Let me check if this triangle is equilateral.

First, compute the lengths of sides AB, AC, and BC.

Starting with AB: distance between (0,0) and (a, $ a^2 / 8 $):

AB = sqrt[ (a - 0)^2 + ( $ a^2 / 8 - 0 $)^2 ] = sqrt[ a^2 + ( $ a^4 $)/64 ]

Similarly, AC is the same since it's symmetric, so AC = AB.

Now, BC is the distance between (a, $ a^2 / 8 $) and (-a, $ a^2 / 8 $). Since they have the same y-coordinate, the distance is just the difference in x-coordinates:

BC = sqrt[ (-a - a)^2 + ( $ a^2 / 8 - a^2 / 8 $)^2 ] = sqrt[ (-2a)^2 + 0 ] = sqrt[ 4a^2 ] = 2|a|.

Since the triangle is equilateral, AB = BC. Therefore:

sqrt[ a^2 + ( $ a^4 $)/64 ] = 2|a|

Let me square both sides to eliminate the square roots:

a^2 + $ a^4 / 64 $ = 4a^2

Bring all terms to one side:

a^2 + $ a^4 / 64 $ - 4a^2 = 0 => -3a^2 + $ a^4 / 64 $ = 0

Multiply both sides by 64 to eliminate denominator:

-192a^2 + a^4 = 0 => a^4 - 192a^2 = 0 => a^2(a^2 - 192) = 0

So, solutions are a^2 = 0 or a^2 = 192.

a^2 = 0 would mean a = 0, which would make points B and C both at (0,0), which coincides with point A, so that's degenerate. Therefore, we discard that solution and take a^2 = 192, so a = sqrt(192) or -sqrt(192). Since we're dealing with coordinates, both are valid, but since we're squaring, it's symmetric.

Therefore, a = sqrt(192) = sqrt(64*3) = 8*sqrt(3). Therefore, coordinates of B and C are (8√3, ( (8√3)^2 ) /8 ) and (-8√3, same y-coordinate).

Calculating the y-coordinate: ( (8√3)^2 ) / 8 = (64 * 3)/8 = (192)/8 = 24. So points B and C are (8√3, 24) and (-8√3, 24).

Now, let's check the length AB. Earlier, we had AB = sqrt[ a^2 + (a^4)/64 ] with a^2 = 192. Let's compute:

a^2 = 192, so a^4 = (192)^2. Therefore:

AB = sqrt(192 + (192^2)/64). Let me compute that:

First, 192 / 64 = 3, so (192^2)/64 = 192 * 3 = 576. Therefore:

AB = sqrt(192 + 576) = sqrt(768). Simplify sqrt(768). 768 = 256 * 3, so sqrt(256*3) = 16√3. Alternatively, check BC: BC was 2|a| = 2*(8√3) = 16√3. So indeed, AB = BC = 16√3, so the triangle is equilateral. So that seems to work. Therefore, the side length is 16√3.

But wait, before finalizing, let me check if this is correct. Let me compute AB again using coordinates. Point A (0,0), point B (8√3, 24). The distance is sqrt[(8√3 - 0)^2 + (24 - 0)^2] = sqrt[ (64 * 3) + 576 ] = sqrt[192 + 576] = sqrt[768] which is indeed 16√3 as above. So that seems correct.

But wait, let me think again. If the triangle is equilateral, not only AB = BC, but also AC = AB, which we already have because of symmetry. So that seems okay. Therefore, is this the answer? 16√3?

Wait, but let me check if there could be another triangle where points B and C are not symmetric with respect to the y-axis? Maybe there's another configuration where the triangle is equilateral but not symmetric. But since the parabola is symmetric and one vertex is at the origin, maybe the other two points are symmetric. But maybe not. However, given that the problem states the triangle is inscribed in the parabola with one vertex at the origin, maybe there is only one such triangle up to symmetry. However, maybe there are multiple triangles. But perhaps we need to confirm.

Alternatively, maybe my assumption of symmetry is leading me to the correct answer. But maybe not. Let me think.

Alternatively, suppose that the triangle is not symmetric. Let me try to think of general coordinates.

Let me suppose that one vertex is at (0,0), and the other two vertices are (x1, y1) and (x2, y2) on the parabola x²=8y. Since they are on the parabola, y1 = x1² /8, y2 = x2² /8.

Since the triangle is equilateral, the distances between each pair of points should be equal. Therefore:

Distance between (0,0) and (x1, y1): sqrt(x1² + y1²) = sqrt(x1² + (x1² /8)^2 )

Distance between (0,0) and (x2, y2): sqrt(x2² + (x2² /8)^2 )

Distance between (x1, y1) and (x2, y2): sqrt( (x1 - x2)^2 + (y1 - y2)^2 )

For the triangle to be equilateral, all three distances must be equal.

Therefore, we need:

sqrt(x1² + (x1² /8)^2 ) = sqrt(x2² + (x2² /8)^2 ) = sqrt( (x1 - x2)^2 + (y1 - y2)^2 )

First, the first two distances are equal. Therefore:

sqrt(x1² + (x1^4)/64 ) = sqrt(x2² + (x2^4)/64 )

Squaring both sides:

x1² + x1^4 /64 = x2² + x2^4 /64

Which implies:

x1^4 /64 + x1² = x2^4 /64 + x2²

Let me rearrange:

(x1^4 - x2^4)/64 + (x1² - x2²) = 0

Factor x1^4 - x2^4 = (x1² - x2²)(x1² + x2²) = (x1 - x2)(x1 + x2)(x1² + x2²)

Therefore:

[(x1 - x2)(x1 + x2)(x1² + x2²)/64] + (x1 - x2)(x1 + x2) = 0

Factor out (x1 - x2)(x1 + x2):

(x1 - x2)(x1 + x2)[ (x1² + x2²)/64 + 1 ] = 0

So, solutions are either x1 = x2, which would mean the two points are the same, which is not allowed, or x1 = -x2, which would be the symmetric case we considered earlier, or the term in brackets equals zero:

(x1² + x2²)/64 + 1 = 0. But since x1² and x2² are non-negative, their sum divided by 64 is non-negative, so adding 1 gives something ≥1 >0. Therefore, no solution here. Therefore, the only possibilities are x1 = x2 (invalid) or x1 = -x2. Therefore, the only non-degenerate case is when the two points are symmetric with respect to the y-axis. Therefore, our initial assumption was correct. Therefore, there is only one such triangle (up to reflection) which is symmetric with respect to y-axis. Therefore, the side length is 16√3. Therefore, is this the answer?

Wait, but let me check again. Wait, the problem says "an equilateral triangle is inscribed in the parabola... such that one of the vertices coincides with the vertex of the parabola." So maybe there's another triangle where the other vertices are not symmetric? Wait, but according to the above, only solutions are symmetric ones. Therefore, perhaps the answer is indeed 16√3. But let me check with coordinates.

But let me think again. Suppose we have triangle with vertices at (0,0), (a, a²/8), (-a, a²/8). Then, the side lengths are AB = AC = sqrt(a² + (a^4)/64), and BC = 2a. Setting them equal, so sqrt(a² + (a^4)/64) = 2a. Then squaring gives a² + a^4 /64 = 4a² => a^4 /64 = 3a² => a^4 = 192a² => a² = 192 (since a ≠0). Therefore, a²=192, so a= sqrt(192)= 8*sqrt(3). Therefore, BC = 2a= 16*sqrt(3). Therefore, the side length is 16√3. Therefore, that's the answer. Seems straightforward.

But let me check the calculation again. Maybe I made a mistake in calculation.

Wait, sqrt(a^2 + (a^4)/64) equals 2a?

Let me square both sides: a² + (a^4)/64 = 4a² => (a^4)/64 = 3a² => multiply both sides by 64: a^4 = 192a² => a^4 - 192a² = 0 => a²(a² - 192) = 0. So yes, that's correct. Therefore, a²=192. So that's correct.

But wait, if I plug in a=8√3, then y-coordinate is a²/8 = (192)/8 = 24. Therefore, points are (8√3,24), (-8√3,24), and (0,0). Therefore, distance between (8√3,24) and (-8√3,24) is 16√3. Distance between (0,0) and (8√3,24): sqrt( (8√3)^2 + 24^2 ) = sqrt( 64*3 + 576 ) = sqrt( 192 + 576 ) = sqrt(768) = sqrt(256*3) = 16√3. Yes, that works. Therefore, seems correct.

But I have a doubt because sometimes problems like this can have other solutions. Let me think again. Suppose there is another point not symmetric. Wait, but according to the earlier analysis, the only way for the distances from the origin to be equal is if points are symmetric. Therefore, there's only one such triangle.

Alternatively, perhaps there's a triangle where the three points are not symmetric? For example, maybe two points on one side of the axis and one on the other? Wait, but with one vertex at the origin, which is on the axis. If the other two vertices are not symmetric, then maybe?

But according to our previous analysis, for the distances from the origin to be equal, points must be symmetric. However, maybe even if they are not symmetric, but with different distances? Wait, no. For the triangle to be equilateral, all sides must be equal. Therefore, in particular, the sides from the origin to each of the other two points must be equal. Therefore, that would require that those two points are equidistant from the origin. Therefore, since they lie on the parabola, that would mean that their distance from origin is same. Which for parabola x²=8y, points (x,y) and (x', y') have same distance from origin if sqrt(x² + y²) = sqrt(x'^2 + y'^2). Which would lead to x² + y² = x'^2 + y'^2. Since y = x²/8, so x² + (x^4)/64 = x'^2 + (x'^4)/64. Which again leads to the same equation as before, leading to x' = -x or x' = something else. Which we saw only possible if x' = -x. Therefore, there are no other solutions.

Therefore, the only possible triangle is the symmetric one. Therefore, answer is 16√3.

But let me check with another approach, maybe using slopes or angles. Let me think.

In an equilateral triangle, the angle between any two sides is 60 degrees. Let me consider vectors or coordinates.

Take points A(0,0), B(a, a²/8), C(-a, a²/8). Then vectors AB and AC can be considered. The angle between AB and AC should be 60 degrees. Wait, but since the triangle is equilateral, all angles are 60 degrees. Wait, but in our case, points A, B, C. Let me check the angle at point A.

Vectors AB and AC. AB is (a, a²/8) and AC is (-a, a²/8). The angle between AB and AC can be found using the dot product:

cos(theta) = (AB · AC) / (|AB| |AC| )

Compute AB · AC = (a)(-a) + (a²/8)(a²/8) = -a² + (a^4)/64

|AB| |AC| = |AB|^2 = a² + (a^4)/64 (since |AB| = |AC|)

Therefore,

cos(theta) = [ -a² + (a^4)/64 ] / [ a² + (a^4)/64 ]

But since theta is 60 degrees, cos(theta) = 0.5.

Therefore,

[ -a² + (a^4)/64 ] / [ a² + (a^4)/64 ] = 1/2

Cross multiplying:

2[ -a² + (a^4)/64 ] = [ a² + (a^4)/64 ]

Expanding:

-2a² + (a^4)/32 = a² + (a^4)/64

Bring all terms to left:

-2a² + (a^4)/32 - a² - (a^4)/64 = 0 => -3a² + (a^4)/64 = 0

Which is the same equation as before: a^4 /64 - 3a² = 0 => a²(a² / 64 - 3 ) = 0 => a² = 0 or a² = 192. So same result. Hence, this confirms the previous calculation. Therefore, the answer is indeed 16√3.

But let me check if there's a different way to approach this problem, perhaps using parametric equations of the parabola.

Parametric equations for parabola x² = 4py is (2pt, pt²). In our case, 4p=8, so p=2, so parametric equations would be (4t, 2t²). Wait, let me check: standard parametric equations for x²=4py is (2pt, pt²). So here, 4p=8, so p=2. Therefore, parametric equations would be (2*2*t, 2*t²) = (4t, 2t²). Alternatively, maybe someone uses different parametrization. Let me check:

If we take parameter t such that x = 2pt, then x² = 4p²t², and since for parabola x²=4py, then 4p²t² = 4py => y = pt². Therefore, yes, parametric equations are (2pt, pt²). Therefore, for x²=8y, 4p=8 => p=2. Therefore, parametric equations are x=4t, y=2t². Therefore, points on parabola can be represented as (4t, 2t²).

Therefore, let me parametrize points B and C as (4t, 2t²) and (4s, 2s²). Then, since one vertex is at (0,0), which is the origin. Then, triangle ABC is equilateral.

So, distances AB, AC, BC must be equal. Let me write equations.

First, AB^2 = (4t - 0)^2 + (2t² - 0)^2 = 16t² + 4t^4.

Similarly, AC^2 = 16s² + 4s^4.

Since AB = AC, then 16t² + 4t^4 = 16s² + 4s^4, which simplifies to 4t^4 + 16t² = 4s^4 + 16s² => t^4 + 4t² = s^4 + 4s². Which is similar to previous equation. As before, this leads to t^2 = s^2 or something else. If t^2 ≠ s^2, then t^4 - s^4 + 4t² - 4s² =0 => (t^2 - s^2)(t^2 + s^2) + 4(t^2 - s^2) = 0 => (t^2 - s^2)(t^2 + s^2 + 4) = 0. Hence, t^2 = s^2 or t^2 + s^2 +4=0. But since t and s are real numbers, t^2 + s^2 +4 cannot be zero. Therefore, t^2 = s^2. Therefore, s = t or s = -t. If s = t, then points B and C are same, which is invalid. Therefore, s = -t. Therefore, points B and C are (4t, 2t²) and (-4t, 2t²). Therefore, same as previous case, except here parameter t instead of a/4. Let me check.

Earlier, we had points (a, a²/8). Let me see if that is same as parametric equations. If we take parametric equations (4t, 2t²). Let me see for x = 4t, then y = 2t². But in our previous case, points are (a, a²/8). Therefore, if we set 4t = a, then t = a/4. Then y = 2*(a/4)^2 = 2*(a²)/16 = a²/8. Therefore, yes, same as previous. Therefore, this parametrization is consistent.

So then, BC distance squared is [ (4t - (-4t))² + (2t² - 2t²)^2 ] = (8t)^2 + 0 = 64t².

Therefore, BC = 8|t|.

Earlier, AB^2 was 16t² + 4t^4. So AB = sqrt(16t² + 4t^4). For equilateral triangle, AB = BC:

sqrt(16t² + 4t^4) = 8|t|. Squaring both sides:

16t² + 4t^4 = 64t² => 4t^4 -48t² = 0 => 4t²(t² - 12) = 0. Therefore, t² = 0 or t² = 12. Again, t²=0 is trivial. Therefore, t²=12 => t=±2√3.

Therefore, t=2√3, then points B is (4*(2√3), 2*(2√3)^2) = (8√3, 2*(12)) = (8√3, 24). Which is same as before. Therefore, BC = 8|t| = 8*(2√3)=16√3. Hence, side length is 16√3. Same result. Therefore, seems consistent.

Therefore, regardless of the approach, I get 16√3. So maybe that is the answer.

But let me check if there's another triangle where maybe the vertex is not on the axis? Wait, no. The problem says one of the vertices coincides with the vertex of the parabola, which is at (0,0). Therefore, the other two vertices are on the parabola, which we found in symmetric positions.

Alternatively, maybe there's a triangle where the vertex at (0,0) is not the apex? Wait, but in an equilateral triangle, all vertices are the same in terms of distances. But since we have one vertex at (0,0), and the other two on the parabola. But according to our previous reasoning, the only way is symmetric. Therefore, I think our answer is correct.

But just to be thorough, let me consider another approach using coordinates and rotation.

In an equilateral triangle, rotating a point by 60 degrees around another point gives the third point. So, suppose we have points A(0,0), B(a, b), and C(c, d). If we rotate point B around point A by 60 degrees, we should get point C or something.

But since we have symmetry, perhaps this would be overcomplicating. Let me try.

Let me suppose that point C is the image of point B rotated 60 degrees around point A. Since we have a symmetric triangle, rotating B around A by 60 degrees would give C. But since the triangle is symmetric with respect to y-axis, rotating B (which is in the right half) by 180 degrees would give C. However, 180 degrees is not 60. But maybe another rotation?

Alternatively, if we have an equilateral triangle with vertex at origin, then the rotation by 60 degrees should take one vertex to another. But this might be a more complex approach.

Let me try. Let me suppose that point B is (a, b) on parabola, then rotating point B around A(0,0) by 60 degrees gives point C. Then, coordinates of C can be calculated.

Rotation matrix for 60 degrees is:

[ cosθ  -sinθ ]
[ sinθ   cosθ ]

Where θ=60°, cosθ=0.5, sinθ=√3/2.

Therefore, if point B is (a, b), then point C is:

( a*0.5 - b*(√3/2), a*(√3/2) + b*0.5 )

But since point C lies on the parabola x²=8y, so coordinates (x,y) must satisfy x²=8y.

Also, since the triangle is equilateral, the distance from A to B equals distance from A to C, which is already satisfied by rotation. Also, distance from B to C should be equal to AB. However, since we rotated, it should be equal. But maybe we can use this condition to find a and b.

So, let's write down the coordinates of point C:

x = 0.5a - (√3/2) b

y = (√3/2)a + 0.5b

Since point C lies on the parabola, we have:

x² = 8y => [0.5a - (√3/2)b]^2 = 8[ (√3/2)a + 0.5b ]

Also, since point B is on the parabola, we have:

a² = 8b => b = a² / 8.

Therefore, substitute b = a² /8 into the equation above.

Let me compute left side:

[0.5a - (√3/2)*(a² /8)]² = [ (a/2) - (√3 a²)/16 ]²

Right side:

8[ (√3 / 2)a + (1/2)(a² / 8) ] = 8[ (√3 a)/2 + a² /16 ] = 8*(√3 a /2) + 8*(a² /16) = 4√3 a + (a²)/2

Therefore, we have:

[ (a/2) - (√3 a²)/16 ]² = 4√3 a + (a²)/2

This seems complex, but let me compute left side.

Let me denote term inside square:

Let me write it as ( (8a - √3 a²)/16 )? Wait:

Wait original term:

a/2 = 8a/16, and (√3 a²)/16, so yes:

Left side: [ (8a - √3 a²)/16 ]² = [ -√3 a² + 8a ]² / 256

Therefore:

[ (-√3 a² + 8a )² ] / 256 = 4√3 a + (a²)/2

Multiply both sides by 256:

(-√3 a² + 8a )² = 256*(4√3 a + a² /2 )

Compute left side: (-√3 a² + 8a )² = (8a - √3 a²)^2 = ( -√3 a² + 8a )² = ( √3 a² - 8a )² = ( (√3 a²)^2 - 2*(√3 a²)*(8a) + (8a)^2 )

Compute term by term:

(√3 a²)^2 = 3a^4

-2*(√3 a²)*(8a) = -16√3 a³

(8a)^2 = 64a²

Therefore, left side is 3a^4 -16√3 a³ +64a².

Right side: 256*(4√3 a + a² /2 ) = 256*( (a²)/2 + 4√3 a ) = 128a² + 1024√3 a

Therefore, equation:

3a^4 -16√3 a³ +64a² = 128a² + 1024√3 a

Bring all terms to left:

3a^4 -16√3 a³ +64a² -128a² -1024√3 a = 0 => 3a^4 -16√3 a³ -64a² -1024√3 a = 0

Factor out a:

a*(3a³ -16√3 a² -64a -1024√3 ) = 0

Solutions are a=0 (trivial), and the cubic equation:

3a³ -16√3 a² -64a -1024√3 = 0

This seems complicated. Maybe I made a mistake in calculation? Let me check.

Wait, let me check the previous steps again.

Starting from:

[ (a/2) - (√3 a²)/16 ]² = 4√3 a + (a²)/2

Let me compute [ (a/2) - (√3 a²)/16 ]² again:

Let me write a/2 as 8a/16, so:

[8a/16 - √3 a² /16]^2 = [ (8a - √3 a²)/16 ]^2 = (8a - √3 a²)^2 / 256. Correct.

Then expanding (8a - √3 a²)^2:

= ( -√3 a² + 8a )^2 = (√3 a² - 8a)^2 = 3a^4 - 2*√3 a²*8a + 64a²

Wait, wait: (m - n)^2 = m² - 2mn + n². Here m = √3 a², n=8a?

Wait no, original term is (8a - √3 a²)^2. Let me write as (-√3 a² + 8a)^2 which is same as (√3 a² - 8a)^2. So:

(√3 a² - 8a)^2 = ( √3 a² )² - 2*(√3 a²)*(8a) + (8a)^2 = 3a^4 - 16√3 a³ + 64a². Yes, that's correct.

Then right side was 256*(4√3 a + a² / 2 ) = 128a² + 1024√3 a. Correct.

So equation is 3a^4 -16√3 a³ +64a² -128a² -1024√3 a =0 => 3a^4 -16√3 a³ -64a² -1024√3 a=0. Correct.

So factoring out a: a(3a³ -16√3 a² -64a -1024√3 )=0. So non-zero solution requires solving:

3a³ -16√3 a² -64a -1024√3 = 0.

This seems very complex. However, we know that from our previous solution, when a = 8√3, let me check if this satisfies the equation.

Let me compute for a = 8√3:

First, compute each term:

a³ = (8√3)^3 = 512*(3√3) = 1536√3

a² = (8√3)^2 = 64*3 = 192

So:

3a³ = 3*1536√3 = 4608√3

-16√3 a² = -16√3 *192 = -3072√3

-64a = -64*(8√3) = -512√3

-1024√3 = -1024√3

Adding all together:

4608√3 -3072√3 -512√3 -1024√3 = (4608 - 3072 - 512 - 1024)√3

Calculate coefficients:

4608 - 3072 = 1536

1536 - 512 = 1024

1024 - 1024 = 0. Therefore, total is 0. Therefore, a=8√3 is a root. Therefore, the cubic equation can be factored as (a - 8√3)(quadratic)=0.

Therefore, if we perform polynomial division or factorization, we can find other roots, but since we already have a valid solution, and given that in previous approaches we found that a=8√3 is the only valid solution (others would be extraneous or complex), this confirms that our previous answer is correct. Therefore, even though this approach is more complicated, it leads us back to the same result.

Therefore, I think confidently that the side length is 16√3. Therefore, the answer is \boxed{16\sqrt{3}}.

**Final Answer**
The side length of the equilateral triangle is \boxed{16\sqrt{3}}.

To find the side length of an **equilateral triangle** inscribed in the parabola $ x^2 = 8y $, such that one of its vertices coincides with the vertex of the parabola, we analyze the geometric and algebraic properties of this configuration.

---

### **Step 1: Understand the Parabola and the Triangle**

The parabola is given by:

$$
x^2 = 8y
$$

This is a standard parabola that opens upwards with vertex at the origin: $ (0, 0) $. Its general form is $ x^2 = 4py $, so here $ p = 2 $, matching the vertex at $ (0, 0) $.

We are told that **one** of the triangle's vertices coincides with the vertex of the parabola, which is at the origin $ A = (0, 0) $, and the other two vertices $ B $ and $ C $ lie on the parabola.

Let’s denote:

- $ A = (0, 0) $
- $ B = (a, b) $
- $ C = (c, d) $

But since the parabola is symmetric about the vertical line through the origin, and the triangle is equilateral, we can exploit this symmetry. **Assume** that $ B $ and $ C $ are symmetric with respect to the y-axis, so:

$$
B = (a, b), \quad C = (-a, b)
$$

From the equation of the parabola, we have:

$$
x^2 = 8y \Rightarrow b = \frac{a^2}{8}
$$

Thus, coordinates are:

- $ B = \left(a, \frac{a^2}{8}\right) $
- $ C = \left(-a, \frac{a^2}{8}\right) $

---

### **Step 2: Compute Side Lengths**

We now compute the distances between the three vertices. Since the triangle is equilateral, all three sides must be equal.

1. **Distance AB**:

$$
AB = \sqrt{(a - 0)^2 + \left(\frac{a^2}{8} - 0\right)^2} = \sqrt{a^2 + \left(\frac{a^2}{8}\right)^2} = \sqrt{a^2 + \frac{a^4}{64}}
$$

2. **Distance AC**:

Same as AB due to symmetry.

3. **Distance BC**:

$$
BC = \sqrt{(a - (-a))^2 + \left(\frac{a^2}{8} - \frac{a^2}{8}\right)^2} = \sqrt{(2a)^2} = 2a
$$

---

### **Step 3: Set Equal Side Lengths and Solve**

Equating AB = BC:

$$
\sqrt{a^2 + \frac{a^4}{64}} = 2a
$$

Squaring both sides:

$$
a^2 + \frac{a^4}{64} = 4a^2
$$

$$
\frac{a^4}{64} = 3a^2
$$

Multiply both sides by 64:

$$
a^4 = 192a^2
$$

$$
a^2(a^2 - 192) = 0
$$

So $ a^2 = 192 $ (discard $ a^2 = 0 $ since it degenerates the triangle).

Now compute the side length $ BC $:

$$
BC = 2a = 2\sqrt{192} = 2 \cdot \sqrt{64 \cdot 3} = 2 \cdot 8\sqrt{3} = 16\sqrt{3}
$$

---

### **Conclusion**

All sides of the triangle (AB, AC, and BC) are equal to $ 16\sqrt{3} $, confirming it's equilateral. Therefore, the side length of the equilateral triangle is:

$$
\boxed{16\sqrt{3}}
$$

[Assistant]
Explanation:
Okay, so I need to find the side length of an equilateral triangle inscribed in the parabola $ x^2 = 8y $, with one vertex at the vertex of the parabola. Let me start by recalling some properties of parabolas and equilateral triangles.

First, the given parabola is $ x^2 = 8y $. This is a standard upward-opening parabola with vertex at the origin (0,0). The general form of such a parabola is $ x^2 = 4py $, where p is the focal length. Comparing, we have 4p = 8, so p = 2. Therefore, the focus is at (0,2), but maybe that's not directly relevant here. However, knowing the vertex is at (0,0) is important because one of the triangle's vertices is there.

Now, the triangle is equilateral, so all sides are equal. Let me denote the three vertices of the triangle as A, B, and C. Given that one of them is the vertex of the parabola, let's say point A is (0,0). Then points B and C are somewhere on the parabola.

So, we need to find coordinates for points B and C on the parabola such that triangle ABC is equilateral. Since the parabola is symmetric about the y-axis, maybe there's some symmetry in the triangle as well. If the triangle is equilateral, maybe points B and C are symmetric with respect to the y-axis? Let me consider that possibility.

Let me assume that points B and C are symmetric across the y-axis. So if point B is (a, b), then point C would be (-a, b). Since both points lie on the parabola $ x^2 = 8y $, substituting, we get for point B: $ a^2 = 8b $, so b = $ a^2 / 8 $.

Therefore, coordinates of B and C are (a, $ a^2 / 8 $) and (-a, $ a^2 / 8 $).

Now, the triangle ABC has vertices at (0,0), (a, $ a^2 / 8 $), and (-a, $ a^2 / 8 $). Let me check if this triangle is equilateral.

First, compute the lengths of sides AB, AC, and BC.

Starting with AB: distance between (0,0) and (a, $ a^2 / 8 $):

AB = sqrt[ (a - 0)^2 + ( $ a^2 / 8 - 0 $)^2 ] = sqrt[ a^2 + ( $ a^4 $)/64 ]

Similarly, AC is the same since it's symmetric, so AC = AB.

Now, BC is the distance between (a, $ a^2 / 8 $) and (-a, $ a^2 / 8 $). Since they have the same y-coordinate, the distance is just the difference in x-coordinates:

BC = sqrt[ (-a - a)^2 + ( $ a^2 / 8 - a^2 / 8 $)^2 ] = sqrt[ (-2a)^2 + 0 ] = sqrt[ 4a^2 ] = 2|a|.

Since the triangle is equilateral, AB = BC. Therefore:

sqrt[ a^2 + ( $ a^4 $)/64 ] = 2|a|

Let me square both sides to eliminate the square roots:

a^2 + $ a^4 / 64 $ = 4a^2

Bring all terms to one side:

a^2 + $ a^4 / 64 $ - 4a^2 = 0 => -3a^2 + $ a^4 / 64 $ = 0

Multiply both sides by 64 to eliminate denominator:

-192a^2 + a^4 = 0 => a^4 - 192a^2 = 0 => a^2(a^2 - 192) = 0

So, solutions are a^2 = 0 or a^2 = 192.

a^2 = 0 would mean a = 0, which would make points B and C both at (0,0), which coincides with point A, so that's degenerate. Therefore, we discard that solution and take a^2 = 192, so a = sqrt(192) or -sqrt(192). Since we're dealing with coordinates, both are valid, but since we're squaring, it's symmetric.

Therefore, a = sqrt(192) = sqrt(64*3) = 8*sqrt(3). Therefore, coordinates of B and C are (8√3, ( (8√3)^2 ) /8 ) and (-8√3, same y-coordinate).

Calculating the y-coordinate: ( (8√3)^2 ) / 8 = (64 * 3)/8 = (192)/8 = 24. So points B and C are (8√3, 24) and (-8√3, 24).

Now, let's check the length AB. Earlier, we had AB = sqrt[ a^2 + (a^4)/64 ] with a^2 = 192. Let's compute:

a^2 = 192, so a^4 = (192)^2. Therefore:

AB = sqrt(192 + (192^2)/64). Let me compute that:

First, 192 / 64 = 3, so (192^2)/64 = 192 * 3 = 576. Therefore:

AB = sqrt(192 + 576) = sqrt(768). Simplify sqrt(768). 768 = 256 * 3, so sqrt(256*3) = 16√3. Alternatively, check BC: BC was 2|a| = 2*(8√3) = 16√3. So indeed, AB = BC = 16√3, so the triangle is equilateral. So that seems to work. Therefore, the side length is 16√3.

But wait, before finalizing, let me check if this is correct. Let me compute AB again using coordinates. Point A (0,0), point B (8√3, 24). The distance is sqrt[(8√3 - 0)^2 + (24 - 0)^2] = sqrt[ (64 * 3) + 576 ] = sqrt[192 + 576] = sqrt[768] which is indeed 16√3 as above. So that seems correct.

But wait, let me think again. If the triangle is equilateral, not only AB = BC, but also AC = AB, which we already have because of symmetry. So that seems okay. Therefore, is this the answer? 16√3?

Wait, but let me check if there could be another triangle where points B and C are not symmetric with respect to the y-axis? Maybe there's another configuration where the triangle is equilateral but not symmetric. But since the parabola is symmetric and one vertex is at the origin, maybe the other two points are symmetric. But maybe not. However, given that the problem states the triangle is inscribed in the parabola with one vertex at the origin, maybe there is only one such triangle up to symmetry. However, maybe there are multiple triangles. But perhaps we need to confirm.

Alternatively, maybe my assumption of symmetry is leading me to the correct answer. But maybe not. Let me think.

Alternatively, suppose that the triangle is not symmetric. Let me try to think of general coordinates.

Let me suppose that one vertex is at (0,0), and the other two vertices are (x1, y1) and (x2, y2) on the parabola x²=8y. Since they are on the parabola, y1 = x1² /8, y2 = x2² /8.

Since the triangle is equilateral, the distances between each pair of points should be equal. Therefore:

Distance between (0,0) and (x1, y1): sqrt(x1² + y1²) = sqrt(x1² + (x1² /8)^2 )

Distance between (0,0) and (x2, y2): sqrt(x2² + (x2² /8)^2 )

Distance between (x1, y1) and (x2, y2): sqrt( (x1 - x2)^2 + (y1 - y2)^2 )

For the triangle to be equilateral, all three distances must be equal.

Therefore, we need:

sqrt(x1² + (x1² /8)^2 ) = sqrt(x2² + (x2² /8)^2 ) = sqrt( (x1 - x2)^2 + (y1 - y2)^2 )

First, the first two distances are equal. Therefore:

sqrt(x1² + (x1^4)/64 ) = sqrt(x2² + (x2^4)/64 )

Squaring both sides:

x1² + x1^4 /64 = x2² + x2^4 /64

Which implies:

x1^4 /64 + x1² = x2^4 /64 + x2²

Let me rearrange:

(x1^4 - x2^4)/64 + (x1² - x2²) = 0

Factor x1^4 - x2^4 = (x1² - x2²)(x1² + x2²) = (x1 - x2)(x1 + x2)(x1² + x2²)

Therefore:

[(x1 - x2)(x1 + x2)(x1² + x2²)/64] + (x1 - x2)(x1 + x2) = 0

Factor out (x1 - x2)(x1 + x2):

(x1 - x2)(x1 + x2)[ (x1² + x2²)/64 + 1 ] = 0

So, solutions are either x1 = x2, which would mean the two points are the same, which is not allowed, or x1 = -x2, which would be the symmetric case we considered earlier, or the term in brackets equals zero:

(x1² + x2²)/64 + 1 = 0. But since x1² and x2² are non-negative, their sum divided by 64 is non-negative, so adding 1 gives something ≥1 >0. Therefore, no solution here. Therefore, the only possibilities are x1 = x2 (invalid) or x1 = -x2. Therefore, the only non-degenerate case is when the two points are symmetric with respect to the y-axis. Therefore, our initial assumption was correct. Therefore, there is only one such triangle (up to reflection) which is symmetric with respect to y-axis. Therefore, the side length is 16√3. Therefore, is this the answer?

Wait, but let me check again. Wait, the problem says "an equilateral triangle is inscribed in the parabola... such that one of the vertices coincides with the vertex of the parabola." So maybe there's another triangle where the other vertices are not symmetric? Wait, but according to the above, only solutions are symmetric ones. Therefore, perhaps the answer is indeed 16√3. But let me check with coordinates.

But let me think again. Suppose we have triangle with vertices at (0,0), (a, a²/8), (-a, a²/8). Then, the side lengths are AB = AC = sqrt(a² + (a^4)/64), and BC = 2a. Setting them equal, so sqrt(a² + (a^4)/64) = 2a. Then squaring gives a² + a^4 /64 = 4a² => a^4 /64 = 3a² => a^4 = 192a² => a² = 192 (since a ≠0). Therefore, a²=192, so a= sqrt(192)= 8*sqrt(3). Therefore, BC = 2a= 16*sqrt(3). Therefore, the side length is 16√3. Therefore, that's the answer. Seems straightforward.

But let me check the calculation again. Maybe I made a mistake in calculation.

Wait, sqrt(a^2 + (a^4)/64) equals 2a?

Let me square both sides: a² + (a^4)/64 = 4a² => (a^4)/64 = 3a² => multiply both sides by 64: a^4 = 192a² => a^4 - 192a² = 0 => a²(a² - 192) = 0. So yes, that's correct. Therefore, a²=192. So that's correct.

But wait, if I plug in a=8√3, then y-coordinate is a²/8 = (192)/8 = 24. Therefore, points are (8√3,24), (-8√3,24), and (0,0). Therefore, distance between (8√3,24) and (-8√3,24) is 16√3. Distance between (0,0) and (8√3,24): sqrt( (8√3)^2 + 24^2 ) = sqrt( 64*3 + 576 ) = sqrt( 192 + 576 ) = sqrt(768) = sqrt(256*3) = 16√3. Yes, that works. Therefore, seems correct.

But I have a doubt because sometimes problems like this can have other solutions. Let me think again. Suppose there is another point not symmetric. Wait, but according to the earlier analysis, the only way for the distances from the origin to be equal is if points are symmetric. Therefore, there's only one such triangle.

Alternatively, perhaps there's a triangle where the three points are not symmetric? For example, maybe two points on one side of the axis and one on the other? Wait, but with one vertex at the origin, which is on the axis. If the other two vertices are not symmetric, then maybe?

But according to our previous analysis, for the distances from the origin to be equal, points must be symmetric. However, maybe even if they are not symmetric, but with different distances? Wait, no. For the triangle to be equilateral, all sides must be equal. Therefore, in particular, the sides from the origin to each of the other two points must be equal. Therefore, that would require that those two points are equidistant from the origin. Therefore, since they lie on the parabola, that would mean that their distance from origin is same. Which for parabola x²=8y, points (x,y) and (x', y') have same distance from origin if sqrt(x² + y²) = sqrt(x'^2 + y'^2). Which would lead to x² + y² = x'^2 + y'^2. Since y = x²/8, so x² + (x^4)/64 = x'^2 + (x'^4)/64. Which again leads to the same equation as before, leading to x' = -x or x' = something else. Which we saw only possible if x' = -x. Therefore, there are no other solutions.

Therefore, the only possible triangle is the symmetric one. Therefore, answer is 16√3.

But let me check with another approach, maybe using slopes or angles. Let me think.

In an equilateral triangle, the angle between any two sides is 60 degrees. Let me consider vectors or coordinates.

Take points A(0,0), B(a, a²/8), C(-a, a²/8). Then vectors AB and AC can be considered. The angle between AB and AC should be 60 degrees. Wait, but since the triangle is equilateral, all angles are 60 degrees. Wait, but in our case, points A, B, C. Let me check the angle at point A.

Vectors AB and AC. AB is (a, a²/8) and AC is (-a, a²/8). The angle between AB and AC can be found using the dot product:

cos(theta) = (AB · AC) / (|AB| |AC| )

Compute AB · AC = (a)(-a) + (a²/8)(a²/8) = -a² + (a^4)/64

|AB| |AC| = |AB|^2 = a² + (a^4)/64 (since |AB| = |AC|)

Therefore,

cos(theta) = [ -a² + (a^4)/64 ] / [ a² + (a^4)/64 ]

But since theta is 60 degrees, cos(theta) = 0.5.

Therefore,

[ -a² + (a^4)/64 ] / [ a² + (a^4)/64 ] = 1/2

Cross multiplying:

2[ -a² + (a^4)/64 ] = [ a² + (a^4)/64 ]

Expanding:

-2a² + (a^4)/32 = a² + (a^4)/64

Bring all terms to left:

-2a² + (a^4)/32 - a² - (a^4)/64 = 0 => -3a² + (a^4)/64 = 0

Which is the same equation as before: a^4 /64 - 3a² = 0 => a²(a² / 64 - 3 ) = 0 => a² = 0 or a² = 192. So same result. Hence, this confirms the previous calculation. Therefore, the answer is indeed 16√3.

But let me check if there's a different way to approach this problem, perhaps using parametric equations of the parabola.

Parametric equations for parabola x² = 4py is (2pt, pt²). In our case, 4p=8, so p=2, so parametric equations would be (4t, 2t²). Wait, let me check: standard parametric equations for x²=4py is (2pt, pt²). So here, 4p=8, so p=2. Therefore, parametric equations would be (2*2*t, 2*t²) = (4t, 2t²). Alternatively, maybe someone uses different parametrization. Let me check:

If we take parameter t such that x = 2pt, then x² = 4p²t², and since for parabola x²=4py, then 4p²t² = 4py => y = pt². Therefore, yes, parametric equations are (2pt, pt²). Therefore, for x²=8y, 4p=8 => p=2. Therefore, parametric equations are x=4t, y=2t². Therefore, points on parabola can be represented as (4t, 2t²).

Therefore, let me parametrize points B and C as (4t, 2t²) and (4s, 2s²). Then, since one vertex is at (0,0), which is the origin. Then, triangle ABC is equilateral.

So, distances AB, AC, BC must be equal. Let me write equations.

First, AB^2 = (4t - 0)^2 + (2t² - 0)^2 = 16t² + 4t^4.

Similarly, AC^2 = 16s² + 4s^4.

Since AB = AC, then 16t² + 4t^4 = 16s² + 4s^4, which simplifies to 4t^4 + 16t² = 4s^4 + 16s² => t^4 + 4t² = s^4 + 4s². Which is similar to previous equation. As before, this leads to t^2 = s^2 or something else. If t^2 ≠ s^2, then t^4 - s^4 + 4t² - 4s² =0 => (t^2 - s^2)(t^2 + s^2) + 4(t^2 - s^2) = 0 => (t^2 - s^2)(t^2 + s^2 + 4) = 0. Hence, t^2 = s^2 or t^2 + s^2 +4=0. But since t and s are real numbers, t^2 + s^2 +4 cannot be zero. Therefore, t^2 = s^2. Therefore, s = t or s = -t. If s = t, then points B and C are same, which is invalid. Therefore, s = -t. Therefore, points B and C are (4t, 2t²) and (-4t, 2t²). Therefore, same as previous case, except here parameter t instead of a/4. Let me check.

Earlier, we had points (a, a²/8). Let me see if that is same as parametric equations. If we take parametric equations (4t, 2t²). Let me see for x = 4t, then y = 2t². But in our previous case, points are (a, a²/8). Therefore, if we set 4t = a, then t = a/4. Then y = 2*(a/4)^2 = 2*(a²)/16 = a²/8. Therefore, yes, same as previous. Therefore, this parametrization is consistent.

So then, BC distance squared is [ (4t - (-4t))² + (2t² - 2t²)^2 ] = (8t)^2 + 0 = 64t².

Therefore, BC = 8|t|.

Earlier, AB^2 was 16t² + 4t^4. So AB = sqrt(16t² + 4t^4). For equilateral triangle, AB = BC:

sqrt(16t² + 4t^4) = 8|t|. Squaring both sides:

16t² + 4t^4 = 64t² => 4t^4 -48t² = 0 => 4t²(t² - 12) = 0. Therefore, t² = 0 or t² = 12. Again, t²=0 is trivial. Therefore, t²=12 => t=±2√3.

Therefore, t=2√3, then points B is (4*(2√3), 2*(2√3)^2) = (8√3, 2*(12)) = (8√3, 24). Which is same as before. Therefore, BC = 8|t| = 8*(2√3)=16√3. Hence, side length is 16√3. Same result. Therefore, seems consistent.

Therefore, regardless of the approach, I get 16√3. So maybe that is the answer.

But let me check if there's another triangle where maybe the vertex is not on the axis? Wait, no. The problem says one of the vertices coincides with the vertex of the parabola, which is at (0,0). Therefore, the other two vertices are on the parabola, which we found in symmetric positions.

Alternatively, maybe there's a triangle where the vertex at (0,0) is not the apex? Wait, but in an equilateral triangle, all vertices are the same in terms of distances. But since we have one vertex at (0,0), and the other two on the parabola. But according to our previous reasoning, the only way is symmetric. Therefore, I think our answer is correct.

But just to be thorough, let me consider another approach using coordinates and rotation.

In an equilateral triangle, rotating a point by 60 degrees around another point gives the third point. So, suppose we have points A(0,0), B(a, b), and C(c, d). If we rotate point B around point A by 60 degrees, we should get point C or something.

But since we have symmetry, perhaps this would be overcomplicating. Let me try.

Let me suppose that point C is the image of point B rotated 60 degrees around point A. Since we have a symmetric triangle, rotating B around A by 60 degrees would give C. But since the triangle is symmetric with respect to y-axis, rotating B (which is in the right half) by 180 degrees would give C. However, 180 degrees is not 60. But maybe another rotation?

Alternatively, if we have an equilateral triangle with vertex at origin, then the rotation by 60 degrees should take one vertex to another. But this might be a more complex approach.

Let me try. Let me suppose that point B is (a, b) on parabola, then rotating point B around A(0,0) by 60 degrees gives point C. Then, coordinates of C can be calculated.

Rotation matrix for 60 degrees is:

[ cosθ  -sinθ ]
[ sinθ   cosθ ]

Where θ=60°, cosθ=0.5, sinθ=√3/2.

Therefore, if point B is (a, b), then point C is:

( a*0.5 - b*(√3/2), a*(√3/2) + b*0.5 )

But since point C lies on the parabola x²=8y, so coordinates (x,y) must satisfy x²=8y.

Also, since the triangle is equilateral, the distance from A to B equals distance from A to C, which is already satisfied by rotation. Also, distance from B to C should be equal to AB. However, since we rotated, it should be equal. But maybe we can use this condition to find a and b.

So, let's write down the coordinates of point C:

x = 0.5a - (√3/2) b

y = (√3/2)a + 0.5b

Since point C lies on the parabola, we have:

x² = 8y => [0.5a - (√3/2)b]^2 = 8[ (√3/2)a + 0.5b ]

Also, since point B is on the parabola, we have:

a² = 8b => b = a² / 8.

Therefore, substitute b = a² /8 into the equation above.

Let me compute left side:

[0.5a - (√3/2)*(a² /8)]² = [ (a/2) - (√3 a²)/16 ]²

Right side:

8[ (√3 / 2)a + (1/2)(a² / 8) ] = 8[ (√3 a)/2 + a² /16 ] = 8*(√3 a /2) + 8*(a² /16) = 4√3 a + (a²)/2

Therefore, we have:

[ (a/2) - (√3 a²)/16 ]² = 4√3 a + (a²)/2

This seems complex, but let me compute left side.

Let me denote term inside square:

Let me write it as ( (8a - √3 a²)/16 )? Wait:

Wait original term:

a/2 = 8a/16, and (√3 a²)/16, so yes:

Left side: [ (8a - √3 a²)/16 ]² = [ -√3 a² + 8a ]² / 256

Therefore:

[ (-√3 a² + 8a )² ] / 256 = 4√3 a + (a²)/2

Multiply both sides by 256:

(-√3 a² + 8a )² = 256*(4√3 a + a² /2 )

Compute left side: (-√3 a² + 8a )² = (8a - √3 a²)^2 = ( -√3 a² + 8a )² = ( √3 a² - 8a )² = ( (√3 a²)^2 - 2*(√3 a²)*(8a) + (8a)^2 )

Compute term by term:

(√3 a²)^2 = 3a^4

-2*(√3 a²)*(8a) = -16√3 a³

(8a)^2 = 64a²

Therefore, left side is 3a^4 -16√3 a³ +64a².

Right side: 256*(4√3 a + a² /2 ) = 256*( (a²)/2 + 4√3 a ) = 128a² + 1024√3 a

Therefore, equation:

3a^4 -16√3 a³ +64a² = 128a² + 1024√3 a

Bring all terms to left:

3a^4 -16√3 a³ +64a² -128a² -1024√3 a = 0 => 3a^4 -16√3 a³ -64a² -1024√3 a = 0

Factor out a:

a*(3a³ -16√3 a² -64a -1024√3 ) = 0

Solutions are a=0 (trivial), and the cubic equation:

3a³ -16√3 a² -64a -1024√3 = 0

This seems complicated. Maybe I made a mistake in calculation? Let me check.

Wait, let me check the previous steps again.

Starting from:

[ (a/2) - (√3 a²)/16 ]² = 4√3 a + (a²)/2

Let me compute [ (a/2) - (√3 a²)/16 ]² again:

Let me write a/2 as 8a/16, so:

[8a/16 - √3 a² /16]^2 = [ (8a - √3 a²)/16 ]^2 = (8a - √3 a²)^2 / 256. Correct.

Then expanding (8a - √3 a²)^2:

= ( -√3 a² + 8a )^2 = (√3 a² - 8a)^2 = 3a^4 - 2*√3 a²*8a + 64a²

Wait, wait: (m - n)^2 = m² - 2mn + n². Here m = √3 a², n=8a?

Wait no, original term is (8a - √3 a²)^2. Let me write as (-√3 a² + 8a)^2 which is same as (√3 a² - 8a)^2. So:

(√3 a² - 8a)^2 = ( √3 a² )² - 2*(√3 a²)*(8a) + (8a)^2 = 3a^4 - 16√3 a³ + 64a². Yes, that's correct.

Then right side was 256*(4√3 a + a² / 2 ) = 128a² + 1024√3 a. Correct.

So equation is 3a^4 -16√3 a³ +64a² -128a² -1024√3 a =0 => 3a^4 -16√3 a³ -64a² -1024√3 a=0. Correct.

So factoring out a: a(3a³ -16√3 a² -64a -1024√3 )=0. So non-zero solution requires solving:

3a³ -16√3 a² -64a -1024√3 = 0.

This seems very complex. However, we know that from our previous solution, when a = 8√3, let me check if this satisfies the equation.

Let me compute for a = 8√3:

First, compute each term:

a³ = (8√3)^3 = 512*(3√3) = 1536√3

a² = (8√3)^2 = 64*3 = 192

So:

3a³ = 3*1536√3 = 4608√3

-16√3 a² = -16√3 *192 = -3072√3

-64a = -64*(8√3) = -512√3

-1024√3 = -1024√3

Adding all together:

4608√3 -3072√3 -512√3 -1024√3 = (4608 - 3072 - 512 - 1024)√3

Calculate coefficients:

4608 - 3072 = 1536

1536 - 512 = 1024

1024 - 1024 = 0. Therefore, total is 0. Therefore, a=8√3 is a root. Therefore, the cubic equation can be factored as (a - 8√3)(quadratic)=0.

Therefore, if we perform polynomial division or factorization, we can find other roots, but since we already have a valid solution, and given that in previous approaches we found that a=8√3 is the only valid solution (others would be extraneous or complex), this confirms that our previous answer is correct. Therefore, even though this approach is more complicated, it leads us back to the same result.

Therefore, I think confidently that the side length is 16√3. Therefore, the answer is \boxed{16\sqrt{3}}.

**Final Answer**
The side length of the equilateral triangle is \boxed{16\sqrt{3}}.

To find the side length of an **equilateral triangle** inscribed in the parabola $ x^2 = 8y $, such that one of its vertices coincides with the vertex of the parabola, we analyze the geometric and algebraic properties of this configuration.

---

### **Step 1: Understand the Parabola and the Triangle**

The parabola is given by:

$$
x^2 = 8y
$$

This is a standard parabola that opens upwards with vertex at the origin: $ (0, 0) $. Its general form is $ x^2 = 4py $, so here $ p = 2 $, matching the vertex at $ (0, 0) $.

We are told that **one** of the triangle's vertices coincides with the vertex of the parabola, which is at the origin $ A = (0, 0) $, and the other two vertices $ B $ and $ C $ lie on the parabola.

Let’s denote:

- $ A = (0, 0) $
- $ B = (a, b) $
- $ C = (c, d) $

But since the parabola is symmetric about the vertical line through the origin, and the triangle is equilateral, we can exploit this symmetry. **Assume** that $ B $ and $ C $ are symmetric with respect to the y-axis, so:

$$
B = (a, b), \quad C = (-a, b)
$$

From the equation of the parabola, we have:

$$
x^2 = 8y \Rightarrow b = \frac{a^2}{8}
$$

Thus, coordinates are:

- $ B = \left(a, \frac{a^2}{8}\right) $
- $ C = \left(-a, \frac{a^2}{8}\right) $

---

### **Step 2: Compute Side Lengths**

We now compute the distances between the three vertices. Since the triangle is equilateral, all three sides must be equal.

1. **Distance AB**:

$$
AB = \sqrt{(a - 0)^2 + \left(\frac{a^2}{8} - 0\right)^2} = \sqrt{a^2 + \left(\frac{a^2}{8}\right)^2} = \sqrt{a^2 + \frac{a^4}{64}}
$$

2. **Distance AC**:

Same as AB due to symmetry.

3. **Distance BC**:

$$
BC = \sqrt{(a - (-a))^2 + \left(\frac{a^2}{8} - \frac{a^2}{8}\right)^2} = \sqrt{(2a)^2} = 2a
$$

---

### **Step 3: Set Equal Side Lengths and Solve**

Equating AB = BC:

$$
\sqrt{a^2 + \frac{a^4}{64}} = 2a
$$

Squaring both sides:

$$
a^2 + \frac{a^4}{64} = 4a^2
$$

$$
\frac{a^4}{64} = 3a^2
$$

Multiply both sides by 64:

$$
a^4 = 192a^2
$$

$$
a^2(a^2 - 192) = 0
$$

So $ a^2 = 192 $ (discard $ a^2 = 0 $ since it degenerates the triangle).

Now compute the side length $ BC $:

$$
BC = 2a = 2\sqrt{192} = 2 \cdot \sqrt{64 \cdot 3} = 2 \cdot 8\sqrt{3} = 16\sqrt{3}
$$

---

### **Conclusion**

All sides of the triangle (AB, AC, and BC) are equal to $ 16\sqrt{3} $, confirming it's equilateral. Therefore, the side length of the equilateral triangle is:

$$
\boxed{16\sqrt{3}}
$$

Solution:
The vertex of the parabola is the origin.  Let $A = \left( a, \frac{a^2}{8} \right)$ be one vertex of the triangle.  Then by symmetry, $B = \left( -a, \frac{a^2}{8} \right)$ is another vertex of the triangle.

[asy]
unitsize(0.2 cm);

real parab (real x) {
  return(x^2/8);
}

pair A, B, O;

A = (8*sqrt(3),24);
B = (-8*sqrt(3),24);
O = (0,0);

draw(O--A--B--cycle);
draw(graph(parab,-15,15));

dot("$A = (a,\frac{a^2}{8})$", A, E);
dot("$B = (-a,\frac{a^2}{8})$", B, W);
dot("$O = (0,0)$", O, S);
[/asy]

Then $AB^2 = (a + a)^2 = 4a^2,$ and
\[OA^2 = a^2 + \frac{a^4}{64}.\]Hence, $4a^2 = a^2 + \frac{a^4}{64}.$  Then $3a^2 = \frac{a^4}{64},$ so $a^2 = 3 \cdot 64,$ which means $a = 8 \sqrt{3}.$

Therefore, the side length of the triangle is $2a = \boxed{16 \sqrt{3}}.$


Explanation:
Okay, so I need to figure out the equation of the set of image points when each point on the circle x² + y² = 25 is reflected over the point (4,1). The final equation should be in the form x² + a y² + b x + c y + d = 0, and I need to find the ordered quadruple (a, b, c, d). Let me start by recalling what reflecting a point over another point means.

Reflection over a point typically means that the point we're reflecting over is the midpoint between the original point and its image. So, if I have a point P(x, y) on the original circle, and its image after reflection is P'(x', y'), then (4,1) should be the midpoint of P and P'.

So, the midpoint formula says that the midpoint M between P(x, y) and P'(x', y') is given by:

M_x = (x + x') / 2 = 4

M_y = (y + y') / 2 = 1

So, solving for x' and y', we can write:

x' = 2*4 - x = 8 - x

y' = 2*1 - y = 2 - y

Therefore, the reflection of any point (x, y) on the original circle over (4,1) is the point (8 - x, 2 - y).

Now, since the original points (x, y) lie on the circle x² + y² = 25, I need to find the equation that describes the set of points (x', y') which are reflections of these original points.

So, substituting x' = 8 - x and y' = 2 - y into the original equation. Wait, but I need to express the equation in terms of x' and y', right? Let me think.

Alternatively, I can express x and y in terms of x' and y'. Since x' = 8 - x, that implies x = 8 - x'. Similarly, y' = 2 - y implies y = 2 - y'.

Therefore, substituting these into the original circle equation:

Original equation: x² + y² = 25

Substitute x = 8 - x', y = 2 - y':

(8 - x')² + (2 - y')² = 25

Expanding this equation should give me the equation in terms of x' and y', which would be the equation for the image points. Then, I can compare it to the given form x² + a y² + b x + c y + d = 0, and find the coefficients.

Let me compute that expansion:

First, expand (8 - x')²:

= 8² - 2*8*x' + x'² = 64 - 16x' + x'²

Then, expand (2 - y')²:

= 2² - 2*2*y' + y'² = 4 - 4y' + y'²

Adding these together:

64 - 16x' + x'² + 4 - 4y' + y'² = 25

Combine like terms:

64 + 4 = 68

So: x'² + y'² - 16x' - 4y' + 68 = 25

Subtract 25 from both sides:

x'² + y'² - 16x' - 4y' + 68 - 25 = 0

Which simplifies to:

x'² + y'² - 16x' - 4y' + 43 = 0

So, if I replace x' with x and y' with y (since the variables are just placeholders), the equation becomes:

x² + y² - 16x - 4y + 43 = 0

Comparing this to the given form x² + a y² + b x + c y + d = 0, I can see that:

a = 1 (since coefficient of y² is 1)

Wait, hold on. Wait, the original equation after substitution is x² + y² - 16x - 4y + 43 = 0. So, in terms of the given form, which is x² + a y² + b x + c y + d = 0, so that would mean that a is 1 (the coefficient of y²), b is -16 (the coefficient of x), c is -4 (coefficient of y), and d is 43.

Wait, but the original equation is x² + y² - 16x - 4y + 43 = 0, which matches exactly the form x² + 1 y² + (-16)x + (-4)y + 43 = 0. Therefore, the ordered quadruple (a, b, c, d) is (1, -16, -4, 43).

But let me double-check my steps to make sure I didn't make any mistakes.

Starting with reflecting the point (x, y) over (4,1) gives (8 - x, 2 - y). Then, substituting into the original equation. Wait, maybe I should check the substitution again.

Original circle: x² + y² = 25. The image points are (x', y') where x' = 8 - x and y' = 2 - y. Therefore, solving for x and y gives x = 8 - x', y = 2 - y'. Therefore substituting into original equation:

(8 - x')² + (2 - y')² = 25. Then expanding:

64 - 16x' + x'² + 4 - 4y' + y'² = 25. Adding 64 + 4 gives 68. Then subtract 25: 68 -25 = 43. So equation is x'² + y'² -16x' -4y' +43=0. So yes, that seems correct.

Therefore, the equation is x² + y² -16x -4y +43=0. So comparing with x² + a y² + b x + c y + d = 0, we have:

a=1, b=-16, c=-4, d=43.

But wait, the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". So the equation we found is x² + y² -16x -4y +43 = 0. Therefore, the coefficients are:

a=1, b=-16, c=-4, d=43. Therefore ordered quadruple is (1, -16, -4, 43). Is there any chance that a is different? Let me check again.

Original equation after substitution is x² + y² -16x -4y +43 = 0. So it's x² + y² + ... So the coefficient of y² is 1, so a is 1. Therefore yes, that seems correct.

But maybe I need to check if there's another way to approach this. For example, maybe the reflection of a circle over a point is another circle. Let me recall that reflecting a circle over a point results in another circle with the same radius, but centered at the reflection of the original center over that point.

Original circle: center at (0, 0), radius 5. Reflecting the center (0,0) over the point (4,1) would give a new center. How?

Using the same midpoint formula: if (4,1) is the midpoint between original center (0,0) and the new center (h, k), then:

( (0 + h)/2 , (0 + k)/2 ) = (4,1). Therefore, h/2 = 4 => h=8, k/2=1 => k=2. So the new center is (8,2), and the radius remains 5. Therefore, the equation of the reflected circle should be (x - 8)^2 + (y - 2)^2 = 25.

Expanding this equation:

x² - 16x + 64 + y² -4y +4 =25.

Combine terms:

x² + y² -16x -4y +64 +4 -25 = 0.

Which is x² + y² -16x -4y + (68 -25)= x² + y² -16x -4y +43 =0. Which matches exactly with the equation I found earlier. Therefore, this confirms that the equation is correct, so the coefficients are indeed a=1, b=-16, c=-4, d=43. Therefore, the ordered quadruple is (1, -16, -4, 43).

Wait, but let me check once again if the question says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". Our equation is x² + y² -16x -4y +43 = 0, which can be written as x² + 1 y² -16x -4y +43 = 0. Therefore, a is 1, b is -16, c is -4, d is 43. Hence, the ordered quadruple is (1, -16, -4, 43).

I think that is correct. Let me check for any possible mistake. For example, sometimes when reflecting, maybe there's a different approach? But no, reflecting over a point is just translating the center, and since the reflection of a circle over a point is another circle with the same radius, centered at the reflection of the original center. Therefore, the process is correct.

Alternatively, maybe I can take a sample point on the original circle, reflect it, and check if it satisfies the equation. For example, take the point (5, 0) on the original circle (since 5² +0²=25). Reflecting over (4,1): x' = 8 -5=3, y'=2 -0=2. So the image point is (3,2). Let me plug into the equation x² + y² -16x -4y +43.

Calculating 3² + 2² -16*3 -4*2 +43 = 9 +4 -48 -8 +43 = 13 -56 +43 = (13 +43) -56 = 56 -56 = 0. Correct.

Another point: (0,5) on original circle. Reflecting over (4,1): x'=8 -0=8, y'=2 -5= -3. So image point (8, -3). Plug into equation: 8² + (-3)^2 -16*8 -4*(-3) +43 = 64 +9 -128 +12 +43. Let's compute:

64 +9 =73; 73 -128 = -55; -55 +12 = -43; -43 +43 = 0. Perfect.

Another test: (-3, 4) on original circle? Wait, (-3)^2 +4^2=9 +16=25. Yes. Reflecting over (4,1): x'=8 -(-3)=11, y'=2 -4= -2. So image point (11, -2). Plug into equation:

11² + (-2)^2 -16*11 -4*(-2) +43 = 121 +4 -176 +8 +43.

Compute step by step:

121 +4=125; 125 -176= -51; -51 +8= -43; -43 +43=0. Correct again.

So, all test points satisfy the equation. Therefore, the equation is correct, so the answer is (a,b,c,d)=(1, -16, -4, 43).

**Final Answer**
The ordered quadruple is \boxed{(1, -16, -4, 43)}.

To solve the problem, we begin by understanding that reflecting every point on a given circle over a specific point results in another circle. Specifically, reflecting the circle $ x^2 + y^2 = 25 $ (which has center at the origin and radius 5) over the point $ (4,1) $, the image will also be a circle.

---

### Step 1: Reflection of Points

Any point $ (x, y) $ on the original circle reflects over the point $ (4,1) $ to produce an image point $ (x', y') $. The reflection over a point means that the original point and its image are symmetric with respect to the reflecting point. This results in:
$$
x' = 8 - x \quad \text{and} \quad y' = 2 - y
$$
Consequently, $ x = 8 - x' $ and $ y = 2 - y' $.

---

### Step 2: Substitution into Original Circle Equation

Since $ x^2 + y^2 = 25 $, we substitute $ x = 8 - x' $ and $ y = 2 - y' $ into this equation:

$$
(8 - x')^2 + (2 - y')^2 = 25
$$

Expanding both terms:

$$
(8 - x')^2 = 64 - 16x' + x'^2 \quad \text{and} \quad (2 - y')^2 = 4 - 4y' + y'^2
$$

Adding these:

$$
64 - 16x' + x'^2 + 4 - 4y' + y'^2 = 25
$$

Simplifying:

$$
x'^2 + y'^2 - 16x' - 4y' + 68 = 25
$$

Subtract 25 from both sides:

$$
x'^2 + y'^2 - 16x' - 4y' + 43 = 0
$$

---

### Step 3: Rewriting the Equation

Replace $ x' $ with $ x $ and $ y' $ with $ y $ (since we're just renaming variables for the final equation), we get:

$$
x^2 + y^2 - 16x - 4y + 43 = 0
$$

Now compare this to the standard form given in the problem:
$$
x^2 + a y^2 + b x + c y + d = 0
$$

Matching the coefficients gives:

- $ a = 1 $
- $ b = -16 $
- $ c = -4 $
- $ d = 43 $

---

### Final Answer

$$
\boxed{(1, -16, -4, 43)}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the equation of the set of image points when each point on the circle x² + y² = 25 is reflected over the point (4,1). The final equation should be in the form x² + a y² + b x + c y + d = 0, and I need to find the ordered quadruple (a, b, c, d). Let me start by recalling what reflecting a point over another point means.

Reflection over a point typically means that the point we're reflecting over is the midpoint between the original point and its image. So, if I have a point P(x, y) on the original circle, and its image after reflection is P'(x', y'), then (4,1) should be the midpoint of P and P'.

So, the midpoint formula says that the midpoint M between P(x, y) and P'(x', y') is given by:

M_x = (x + x') / 2 = 4

M_y = (y + y') / 2 = 1

So, solving for x' and y', we can write:

x' = 2*4 - x = 8 - x

y' = 2*1 - y = 2 - y

Therefore, the reflection of any point (x, y) on the original circle over (4,1) is the point (8 - x, 2 - y).

Now, since the original points (x, y) lie on the circle x² + y² = 25, I need to find the equation that describes the set of points (x', y') which are reflections of these original points.

So, substituting x' = 8 - x and y' = 2 - y into the original equation. Wait, but I need to express the equation in terms of x' and y', right? Let me think.

Alternatively, I can express x and y in terms of x' and y'. Since x' = 8 - x, that implies x = 8 - x'. Similarly, y' = 2 - y implies y = 2 - y'.

Therefore, substituting these into the original circle equation:

Original equation: x² + y² = 25

Substitute x = 8 - x', y = 2 - y':

(8 - x')² + (2 - y')² = 25

Expanding this equation should give me the equation in terms of x' and y', which would be the equation for the image points. Then, I can compare it to the given form x² + a y² + b x + c y + d = 0, and find the coefficients.

Let me compute that expansion:

First, expand (8 - x')²:

= 8² - 2*8*x' + x'² = 64 - 16x' + x'²

Then, expand (2 - y')²:

= 2² - 2*2*y' + y'² = 4 - 4y' + y'²

Adding these together:

64 - 16x' + x'² + 4 - 4y' + y'² = 25

Combine like terms:

64 + 4 = 68

So: x'² + y'² - 16x' - 4y' + 68 = 25

Subtract 25 from both sides:

x'² + y'² - 16x' - 4y' + 68 - 25 = 0

Which simplifies to:

x'² + y'² - 16x' - 4y' + 43 = 0

So, if I replace x' with x and y' with y (since the variables are just placeholders), the equation becomes:

x² + y² - 16x - 4y + 43 = 0

Comparing this to the given form x² + a y² + b x + c y + d = 0, I can see that:

a = 1 (since coefficient of y² is 1)

Wait, hold on. Wait, the original equation after substitution is x² + y² - 16x - 4y + 43 = 0. So, in terms of the given form, which is x² + a y² + b x + c y + d = 0, so that would mean that a is 1 (the coefficient of y²), b is -16 (the coefficient of x), c is -4 (coefficient of y), and d is 43.

Wait, but the original equation is x² + y² - 16x - 4y + 43 = 0, which matches exactly the form x² + 1 y² + (-16)x + (-4)y + 43 = 0. Therefore, the ordered quadruple (a, b, c, d) is (1, -16, -4, 43).

But let me double-check my steps to make sure I didn't make any mistakes.

Starting with reflecting the point (x, y) over (4,1) gives (8 - x, 2 - y). Then, substituting into the original equation. Wait, maybe I should check the substitution again.

Original circle: x² + y² = 25. The image points are (x', y') where x' = 8 - x and y' = 2 - y. Therefore, solving for x and y gives x = 8 - x', y = 2 - y'. Therefore substituting into original equation:

(8 - x')² + (2 - y')² = 25. Then expanding:

64 - 16x' + x'² + 4 - 4y' + y'² = 25. Adding 64 + 4 gives 68. Then subtract 25: 68 -25 = 43. So equation is x'² + y'² -16x' -4y' +43=0. So yes, that seems correct.

Therefore, the equation is x² + y² -16x -4y +43=0. So comparing with x² + a y² + b x + c y + d = 0, we have:

a=1, b=-16, c=-4, d=43.

But wait, the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". So the equation we found is x² + y² -16x -4y +43 = 0. Therefore, the coefficients are:

a=1, b=-16, c=-4, d=43. Therefore ordered quadruple is (1, -16, -4, 43). Is there any chance that a is different? Let me check again.

Original equation after substitution is x² + y² -16x -4y +43 = 0. So it's x² + y² + ... So the coefficient of y² is 1, so a is 1. Therefore yes, that seems correct.

But maybe I need to check if there's another way to approach this. For example, maybe the reflection of a circle over a point is another circle. Let me recall that reflecting a circle over a point results in another circle with the same radius, but centered at the reflection of the original center over that point.

Original circle: center at (0, 0), radius 5. Reflecting the center (0,0) over the point (4,1) would give a new center. How?

Using the same midpoint formula: if (4,1) is the midpoint between original center (0,0) and the new center (h, k), then:

( (0 + h)/2 , (0 + k)/2 ) = (4,1). Therefore, h/2 = 4 => h=8, k/2=1 => k=2. So the new center is (8,2), and the radius remains 5. Therefore, the equation of the reflected circle should be (x - 8)^2 + (y - 2)^2 = 25.

Expanding this equation:

x² - 16x + 64 + y² -4y +4 =25.

Combine terms:

x² + y² -16x -4y +64 +4 -25 = 0.

Which is x² + y² -16x -4y + (68 -25)= x² + y² -16x -4y +43 =0. Which matches exactly with the equation I found earlier. Therefore, this confirms that the equation is correct, so the coefficients are indeed a=1, b=-16, c=-4, d=43. Therefore, the ordered quadruple is (1, -16, -4, 43).

Wait, but let me check once again if the question says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". Our equation is x² + y² -16x -4y +43 = 0, which can be written as x² + 1 y² -16x -4y +43 = 0. Therefore, a is 1, b is -16, c is -4, d is 43. Hence, the ordered quadruple is (1, -16, -4, 43).

I think that is correct. Let me check for any possible mistake. For example, sometimes when reflecting, maybe there's a different approach? But no, reflecting over a point is just translating the center, and since the reflection of a circle over a point is another circle with the same radius, centered at the reflection of the original center. Therefore, the process is correct.

Alternatively, maybe I can take a sample point on the original circle, reflect it, and check if it satisfies the equation. For example, take the point (5, 0) on the original circle (since 5² +0²=25). Reflecting over (4,1): x' = 8 -5=3, y'=2 -0=2. So the image point is (3,2). Let me plug into the equation x² + y² -16x -4y +43.

Calculating 3² + 2² -16*3 -4*2 +43 = 9 +4 -48 -8 +43 = 13 -56 +43 = (13 +43) -56 = 56 -56 = 0. Correct.

Another point: (0,5) on original circle. Reflecting over (4,1): x'=8 -0=8, y'=2 -5= -3. So image point (8, -3). Plug into equation: 8² + (-3)^2 -16*8 -4*(-3) +43 = 64 +9 -128 +12 +43. Let's compute:

64 +9 =73; 73 -128 = -55; -55 +12 = -43; -43 +43 = 0. Perfect.

Another test: (-3, 4) on original circle? Wait, (-3)^2 +4^2=9 +16=25. Yes. Reflecting over (4,1): x'=8 -(-3)=11, y'=2 -4= -2. So image point (11, -2). Plug into equation:

11² + (-2)^2 -16*11 -4*(-2) +43 = 121 +4 -176 +8 +43.

Compute step by step:

121 +4=125; 125 -176= -51; -51 +8= -43; -43 +43=0. Correct again.

So, all test points satisfy the equation. Therefore, the equation is correct, so the answer is (a,b,c,d)=(1, -16, -4, 43).

**Final Answer**
The ordered quadruple is \boxed{(1, -16, -4, 43)}.

To solve the problem, we begin by understanding that reflecting every point on a given circle over a specific point results in another circle. Specifically, reflecting the circle $ x^2 + y^2 = 25 $ (which has center at the origin and radius 5) over the point $ (4,1) $, the image will also be a circle.

---

### Step 1: Reflection of Points

Any point $ (x, y) $ on the original circle reflects over the point $ (4,1) $ to produce an image point $ (x', y') $. The reflection over a point means that the original point and its image are symmetric with respect to the reflecting point. This results in:
$$
x' = 8 - x \quad \text{and} \quad y' = 2 - y
$$
Consequently, $ x = 8 - x' $ and $ y = 2 - y' $.

---

### Step 2: Substitution into Original Circle Equation

Since $ x^2 + y^2 = 25 $, we substitute $ x = 8 - x' $ and $ y = 2 - y' $ into this equation:

$$
(8 - x')^2 + (2 - y')^2 = 25
$$

Expanding both terms:

$$
(8 - x')^2 = 64 - 16x' + x'^2 \quad \text{and} \quad (2 - y')^2 = 4 - 4y' + y'^2
$$

Adding these:

$$
64 - 16x' + x'^2 + 4 - 4y' + y'^2 = 25
$$

Simplifying:

$$
x'^2 + y'^2 - 16x' - 4y' + 68 = 25
$$

Subtract 25 from both sides:

$$
x'^2 + y'^2 - 16x' - 4y' + 43 = 0
$$

---

### Step 3: Rewriting the Equation

Replace $ x' $ with $ x $ and $ y' $ with $ y $ (since we're just renaming variables for the final equation), we get:

$$
x^2 + y^2 - 16x - 4y + 43 = 0
$$

Now compare this to the standard form given in the problem:
$$
x^2 + a y^2 + b x + c y + d = 0
$$

Matching the coefficients gives:

- $ a = 1 $
- $ b = -16 $
- $ c = -4 $
- $ d = 43 $

---

### Final Answer

$$
\boxed{(1, -16, -4, 43)}
$$

Solution:
The center of the original circle is $(0,0).$  The reflection of the point $(0,0)$ in the point $(4,1)$ is $(8,2),$ so the equation of the new circle is
\[(x - 8)^2 + (y - 2)^2 = 25.\]This simplifies to $x^2 + y^2 - 16x - 4y + 43 = 0.$  Hence, $(a,b,c,d) = \boxed{(1,-16,-4,43)}.$


Explanation:
Okay, so I need to find the modulo 7 remainder of the sum of all the odd numbers from 1 up to 199. Let me think about how to approach this.

First, let me recall that the sum of the first n odd numbers is n². Wait, is that right? Yeah, I think that's a formula I remember. Like, 1 + 3 = 4, which is 2², then 1 + 3 + 5 = 9, which is 3², so yeah, the sum of the first n odd numbers is n squared. So if I can figure out how many terms there are in this sequence from 1 to 199, then I can square that number and then take modulo 7.

But wait, let me check if that applies here. The problem says "1 + 3 + 5 + 7 + 9 + ... + 195 + 197 + 199". So that is all the odd numbers starting at 1 and ending at 199. So first, how many terms are there?

To find the number of terms in an arithmetic sequence, the formula is ((last term - first term)/common difference) + 1. In this case, the first term a₁ is 1, the last term a_n is 199, and the common difference d is 2. So plugging into the formula:

n = ((199 - 1)/2) + 1 = (198/2) + 1 = 99 + 1 = 100. So there are 100 terms. Therefore, the sum should be 100² = 10,000. Wait, so the sum is 10,000? Then the question is asking for 10,000 mod 7. Is that right? Let me verify this step by step because sometimes I might mix up formulas.

Alternatively, maybe I should verify the sum another way. Let me check with a small example. If I take the first 3 odd numbers, 1 + 3 + 5 = 9, which is 3². If I take the first 4, 1 + 3 + 5 + 7 = 16, which is 4². So yeah, seems like the formula holds. Therefore, for n terms, sum is n². Therefore, since there are 100 terms, sum is 100² = 10,000. So then, compute 10,000 mod 7.

But maybe I should verify the number of terms again. Let me do it again. The sequence starts at 1, ends at 199, common difference 2. So the nth term is given by a_n = a₁ + (n - 1)d. So 199 = 1 + (n - 1)*2. Subtract 1: 198 = (n - 1)*2. Divide by 2: 99 = n - 1. Therefore, n = 100. Yes, so that's correct. Therefore, 100 terms. So the sum is 100² = 10,000.

Now, compute 10,000 mod 7. How do I do that? Well, maybe divide 10,000 by 7 and find the remainder. Alternatively, find a pattern or simplify the calculation.

Alternatively, since 10,000 is 10^4, maybe I can compute 10 mod 7 first, then raise it to the 4th power? Let's try that.

10 divided by 7 is 1 with remainder 3, so 10 ≡ 3 mod 7. Therefore, 10^4 ≡ 3^4 mod 7. Compute 3^4: 3*3=9, 9*3=27, 27*3=81. So 3^4 is 81. Then, 81 divided by 7: 7*11=77, so 81-77=4. Therefore, 81 ≡ 4 mod 7. Therefore, 10^4 ≡ 4 mod 7. Therefore, 10,000 mod 7 is 4. Therefore, the answer is 4? Wait, but let me check this again.

Alternatively, maybe I can compute 10,000 divided by 7 step by step. Let's see. 7*1428 = 9996. Then 10,000 - 9996 = 4. Therefore, remainder is 4. Yep, same result. So that would mean the answer is 4.

But let me think again. Maybe there's a mistake in assuming that the sum is 100²? Let me check with another approach. For example, compute the sum using the formula for arithmetic series.

The sum of an arithmetic series is (number of terms)/2 * (first term + last term). So here, number of terms is 100, first term is 1, last term is 199. Therefore, sum is (100/2)*(1 + 199) = 50*200 = 10,000. Yep, same result. So that's correct. Therefore, the sum is indeed 10,000, which mod 7 is 4. Therefore, the answer is 4.

But wait, maybe I should check the modulo 7 calculation another way? Let me compute 10,000 divided by 7. Let's do it step by step.

We know that 7*1000 = 7000. Then 10,000 - 7000 = 3000.

Now, 7*400 = 2800. So 3000 - 2800 = 200.

So we have 7*(1000 + 400) = 7*1400 = 9800. Then 10,000 - 9800 = 200.

Now, compute 200 divided by 7. 7*28=196. Therefore, 200 - 196=4. Therefore, total remainder is 4. Yep, same answer.

Alternatively, maybe compute 10,000 mod 7 by breaking it into smaller parts. For example, 10,000 = 10*1000. We can compute 10 mod 7 = 3, and 1000 mod 7. Let me compute 1000 mod 7.

Well, 7*142 = 994. Therefore, 1000 - 994 = 6. Therefore, 1000 ≡ 6 mod 7. Therefore, 10*1000 ≡ 3*6 = 18 ≡ 18 - 14 = 4 mod 7. Yep, same result.

Therefore, regardless of the method, the remainder is 4. Therefore, the answer is 4.

Wait, but maybe I made a mistake in the number of terms? Let me check again. The last term is 199. So starting from 1, each term increases by 2. So the number of terms is ((199 - 1)/2) + 1. Which is (198 / 2) + 1 = 99 + 1 = 100. That's correct. Let me check with a smaller example. Suppose I have numbers up to 5. So 1, 3, 5. Number of terms is ((5 - 1)/2) + 1 = 2 + 1 = 3. Correct. So same formula applies here.

Alternatively, maybe check with 199. Let me think. If I have an odd number, say 199, then the number of terms is (199 + 1)/2 = 200/2 = 100. Wait, that's another formula. Since the nth odd number is 2n - 1. Therefore, solving 2n - 1 = 199 gives 2n = 200, so n = 100. So that's another way to get the number of terms. So that also gives n = 100. Therefore, that's correct.

Therefore, all steps seem correct, so the answer is 4. Therefore, I think the modulo 7 remainder is 4.

But just to make sure, maybe I can compute the sum modulo 7 directly, without calculating the entire sum? Let me try that approach as a cross-check.

So the sum is 1 + 3 + 5 + ... + 199. Let me compute each term modulo 7, then add them up, and take modulo 7 again.

But that might take a while, but maybe we can find a pattern.

First, note that the sequence of odd numbers modulo 7 cycles every certain number of terms. Since the common difference is 2, and modulus is 7, the residues will cycle with period equal to the least common multiple of the difference and modulus divided by gcd(d, modulus). Wait, maybe I need to recall the concept.

Alternatively, since we're adding numbers with difference 2 modulo 7. Let me think.

The sequence is 1, 3, 5, 7, 9, 11, 13, ..., 199. Modulo 7, each term can be represented as follows.

But note that 7 ≡ 0 mod 7, 9 ≡ 2 mod 7, 11 ≡ 4 mod 7, 13 ≡ 6 mod 7, 15 ≡ 1 mod 7, 17 ≡ 3 mod 7, 19 ≡ 5 mod 7, 21 ≡ 0 mod 7, and so on. So the residues cycle every 7 terms? Let me list them:

Term 1: 1 mod 7 = 1

Term 2: 3 mod 7 = 3

Term 3: 5 mod 7 = 5

Term 4: 7 mod 7 = 0

Term 5: 9 mod 7 = 2

Term 6: 11 mod 7 = 4

Term 7: 13 mod 7 = 6

Term 8: 15 mod 7 = 1

Term 9: 17 mod 7 = 3

Term 10: 19 mod 7 = 5

Term 11: 21 mod 7 = 0

... and so on.

So the residues cycle every 7 terms: [1, 3, 5, 0, 2, 4, 6], then repeats again [1, 3, 5, 0, 2, 4, 6], etc.

So the cycle length is 7. Therefore, each cycle of 7 terms contributes 1 + 3 + 5 + 0 + 2 + 4 + 6 = 21.

So sum modulo 7 of each cycle is 21 mod 7 = 0. So each full cycle contributes 0 mod 7. Then, if the total number of terms is a multiple of 7, the total sum mod 7 is 0.

But in our case, total number of terms is 100. Let me compute how many full cycles there are and then the remaining terms.

Number of full cycles = floor(100 / 7). Let me compute 7*14 = 98, so 14 full cycles, which account for 98 terms, and then 100 - 98 = 2 remaining terms.

Each full cycle contributes 21, so total contribution from full cycles is 14*21. Then the remaining 2 terms are term 99 and term 100.

But wait, term 99: Let me figure out what term 99 corresponds to in the original sequence. Since the cycle repeats every 7 terms, term 99 is equivalent to term (99 mod 7). Let's compute 99 divided by 7: 7*14 = 98, so 99 is 98 +1, so term 99 is equivalent to term 1 in the cycle. Similarly, term 100 is equivalent to term 2 in the cycle.

Therefore, the remaining two terms are the first two terms of the cycle: 1 and 3.

Therefore, total sum mod 7 would be (14*21 + 1 + 3) mod 7.

But since 21 mod 7 is 0, 14*21 mod 7 is 0. Therefore, the total sum mod 7 is (0 + 1 + 3) mod 7 = 4 mod 7. Which matches the previous result! So that's reassuring.

Therefore, regardless of the method, the remainder is 4.

Alternatively, maybe I can check with another approach. Let me compute the sum modulo 7 directly using the formula sum = 100² mod 7.

So 100 mod 7: Let's compute that. 7*14=98, so 100 - 98 = 2, so 100 ≡ 2 mod 7. Therefore, 100² ≡ 2² = 4 mod 7. Which again gives 4.

So this third method also gives 4.

Therefore, all methods confirm that the remainder is 4.

Therefore, I think confidently that the answer is 4.

Just to recap:

Method 1: Sum = n² = 100² = 10,000. 10,000 mod 7 = 4.

Method 2: Sum using arithmetic series formula (n/2)(first + last) = 100/2*(1 + 199) = 50*200 = 10,000. 10,000 mod 7 = 4.

Method 3: Direct modulo computation by breaking down the sum into cycles of residues mod 7, which also gave 4.

Therefore, answer is 4.

I think that's thorough. I don't see any mistakes in my reasoning. So I think the answer is 4.

**Final Answer**
\boxed{4}

To solve the problem, we are asked to find the **modulo 7 remainder** of the sum:

$$
1 + 3 + 5 + 7 + \dots + 195 + 197 + 199
$$

---

### Step 1: Determine the Number of Terms

This is an arithmetic sequence of **odd numbers** starting from 1 and ending at 199, with common difference 2.

We can use the formula for the nth term of an arithmetic sequence:

$$
a_n = a_1 + (n-1)d
$$

Here, $ a_1 = 1 $, $ a_n = 199 $, $ d = 2 $, so:

$$
199 = 1 + (n - 1) \cdot 2 \Rightarrow 198 = 2(n - 1) \Rightarrow n - 1 = 99 \Rightarrow n = 100
$$

So, there are **100 terms** in the sequence.

---

### Step 2: Sum of the Sequence

There is a known formula for the sum of the first $ n $ odd numbers:

$$
\text{Sum} = n^2
$$

So, for $ n = 100 $:

$$
\text{Sum} = 100^2 = 10,000
$$

---

### Step 3: Compute $ 10,000 \mod 7 $

Now, we compute the **modulo 7** remainder of 10,000.

We can simplify by computing $ 10,000 \mod 7 $:

We know:

$$
10 \equiv 3 \mod 7 \Rightarrow 10 \equiv 3 \mod 7
\Rightarrow 10^4 \equiv 3^4 \mod 7
$$

Compute:

$$
3^4 = 81
\Rightarrow 81 \mod 7 = 4
$$

Therefore:

$$
10,000 \equiv 10^4 \equiv 3^4 \equiv 81 \equiv 4 \mod 7
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to find the modulo 7 remainder of the sum of all the odd numbers from 1 up to 199. Let me think about how to approach this.

First, let me recall that the sum of the first n odd numbers is n². Wait, is that right? Yeah, I think that's a formula I remember. Like, 1 + 3 = 4, which is 2², then 1 + 3 + 5 = 9, which is 3², so yeah, the sum of the first n odd numbers is n squared. So if I can figure out how many terms there are in this sequence from 1 to 199, then I can square that number and then take modulo 7.

But wait, let me check if that applies here. The problem says "1 + 3 + 5 + 7 + 9 + ... + 195 + 197 + 199". So that is all the odd numbers starting at 1 and ending at 199. So first, how many terms are there?

To find the number of terms in an arithmetic sequence, the formula is ((last term - first term)/common difference) + 1. In this case, the first term a₁ is 1, the last term a_n is 199, and the common difference d is 2. So plugging into the formula:

n = ((199 - 1)/2) + 1 = (198/2) + 1 = 99 + 1 = 100. So there are 100 terms. Therefore, the sum should be 100² = 10,000. Wait, so the sum is 10,000? Then the question is asking for 10,000 mod 7. Is that right? Let me verify this step by step because sometimes I might mix up formulas.

Alternatively, maybe I should verify the sum another way. Let me check with a small example. If I take the first 3 odd numbers, 1 + 3 + 5 = 9, which is 3². If I take the first 4, 1 + 3 + 5 + 7 = 16, which is 4². So yeah, seems like the formula holds. Therefore, for n terms, sum is n². Therefore, since there are 100 terms, sum is 100² = 10,000. So then, compute 10,000 mod 7.

But maybe I should verify the number of terms again. Let me do it again. The sequence starts at 1, ends at 199, common difference 2. So the nth term is given by a_n = a₁ + (n - 1)d. So 199 = 1 + (n - 1)*2. Subtract 1: 198 = (n - 1)*2. Divide by 2: 99 = n - 1. Therefore, n = 100. Yes, so that's correct. Therefore, 100 terms. So the sum is 100² = 10,000.

Now, compute 10,000 mod 7. How do I do that? Well, maybe divide 10,000 by 7 and find the remainder. Alternatively, find a pattern or simplify the calculation.

Alternatively, since 10,000 is 10^4, maybe I can compute 10 mod 7 first, then raise it to the 4th power? Let's try that.

10 divided by 7 is 1 with remainder 3, so 10 ≡ 3 mod 7. Therefore, 10^4 ≡ 3^4 mod 7. Compute 3^4: 3*3=9, 9*3=27, 27*3=81. So 3^4 is 81. Then, 81 divided by 7: 7*11=77, so 81-77=4. Therefore, 81 ≡ 4 mod 7. Therefore, 10^4 ≡ 4 mod 7. Therefore, 10,000 mod 7 is 4. Therefore, the answer is 4? Wait, but let me check this again.

Alternatively, maybe I can compute 10,000 divided by 7 step by step. Let's see. 7*1428 = 9996. Then 10,000 - 9996 = 4. Therefore, remainder is 4. Yep, same result. So that would mean the answer is 4.

But let me think again. Maybe there's a mistake in assuming that the sum is 100²? Let me check with another approach. For example, compute the sum using the formula for arithmetic series.

The sum of an arithmetic series is (number of terms)/2 * (first term + last term). So here, number of terms is 100, first term is 1, last term is 199. Therefore, sum is (100/2)*(1 + 199) = 50*200 = 10,000. Yep, same result. So that's correct. Therefore, the sum is indeed 10,000, which mod 7 is 4. Therefore, the answer is 4.

But wait, maybe I should check the modulo 7 calculation another way? Let me compute 10,000 divided by 7. Let's do it step by step.

We know that 7*1000 = 7000. Then 10,000 - 7000 = 3000.

Now, 7*400 = 2800. So 3000 - 2800 = 200.

So we have 7*(1000 + 400) = 7*1400 = 9800. Then 10,000 - 9800 = 200.

Now, compute 200 divided by 7. 7*28=196. Therefore, 200 - 196=4. Therefore, total remainder is 4. Yep, same answer.

Alternatively, maybe compute 10,000 mod 7 by breaking it into smaller parts. For example, 10,000 = 10*1000. We can compute 10 mod 7 = 3, and 1000 mod 7. Let me compute 1000 mod 7.

Well, 7*142 = 994. Therefore, 1000 - 994 = 6. Therefore, 1000 ≡ 6 mod 7. Therefore, 10*1000 ≡ 3*6 = 18 ≡ 18 - 14 = 4 mod 7. Yep, same result.

Therefore, regardless of the method, the remainder is 4. Therefore, the answer is 4.

Wait, but maybe I made a mistake in the number of terms? Let me check again. The last term is 199. So starting from 1, each term increases by 2. So the number of terms is ((199 - 1)/2) + 1. Which is (198 / 2) + 1 = 99 + 1 = 100. That's correct. Let me check with a smaller example. Suppose I have numbers up to 5. So 1, 3, 5. Number of terms is ((5 - 1)/2) + 1 = 2 + 1 = 3. Correct. So same formula applies here.

Alternatively, maybe check with 199. Let me think. If I have an odd number, say 199, then the number of terms is (199 + 1)/2 = 200/2 = 100. Wait, that's another formula. Since the nth odd number is 2n - 1. Therefore, solving 2n - 1 = 199 gives 2n = 200, so n = 100. So that's another way to get the number of terms. So that also gives n = 100. Therefore, that's correct.

Therefore, all steps seem correct, so the answer is 4. Therefore, I think the modulo 7 remainder is 4.

But just to make sure, maybe I can compute the sum modulo 7 directly, without calculating the entire sum? Let me try that approach as a cross-check.

So the sum is 1 + 3 + 5 + ... + 199. Let me compute each term modulo 7, then add them up, and take modulo 7 again.

But that might take a while, but maybe we can find a pattern.

First, note that the sequence of odd numbers modulo 7 cycles every certain number of terms. Since the common difference is 2, and modulus is 7, the residues will cycle with period equal to the least common multiple of the difference and modulus divided by gcd(d, modulus). Wait, maybe I need to recall the concept.

Alternatively, since we're adding numbers with difference 2 modulo 7. Let me think.

The sequence is 1, 3, 5, 7, 9, 11, 13, ..., 199. Modulo 7, each term can be represented as follows.

But note that 7 ≡ 0 mod 7, 9 ≡ 2 mod 7, 11 ≡ 4 mod 7, 13 ≡ 6 mod 7, 15 ≡ 1 mod 7, 17 ≡ 3 mod 7, 19 ≡ 5 mod 7, 21 ≡ 0 mod 7, and so on. So the residues cycle every 7 terms? Let me list them:

Term 1: 1 mod 7 = 1

Term 2: 3 mod 7 = 3

Term 3: 5 mod 7 = 5

Term 4: 7 mod 7 = 0

Term 5: 9 mod 7 = 2

Term 6: 11 mod 7 = 4

Term 7: 13 mod 7 = 6

Term 8: 15 mod 7 = 1

Term 9: 17 mod 7 = 3

Term 10: 19 mod 7 = 5

Term 11: 21 mod 7 = 0

... and so on.

So the residues cycle every 7 terms: [1, 3, 5, 0, 2, 4, 6], then repeats again [1, 3, 5, 0, 2, 4, 6], etc.

So the cycle length is 7. Therefore, each cycle of 7 terms contributes 1 + 3 + 5 + 0 + 2 + 4 + 6 = 21.

So sum modulo 7 of each cycle is 21 mod 7 = 0. So each full cycle contributes 0 mod 7. Then, if the total number of terms is a multiple of 7, the total sum mod 7 is 0.

But in our case, total number of terms is 100. Let me compute how many full cycles there are and then the remaining terms.

Number of full cycles = floor(100 / 7). Let me compute 7*14 = 98, so 14 full cycles, which account for 98 terms, and then 100 - 98 = 2 remaining terms.

Each full cycle contributes 21, so total contribution from full cycles is 14*21. Then the remaining 2 terms are term 99 and term 100.

But wait, term 99: Let me figure out what term 99 corresponds to in the original sequence. Since the cycle repeats every 7 terms, term 99 is equivalent to term (99 mod 7). Let's compute 99 divided by 7: 7*14 = 98, so 99 is 98 +1, so term 99 is equivalent to term 1 in the cycle. Similarly, term 100 is equivalent to term 2 in the cycle.

Therefore, the remaining two terms are the first two terms of the cycle: 1 and 3.

Therefore, total sum mod 7 would be (14*21 + 1 + 3) mod 7.

But since 21 mod 7 is 0, 14*21 mod 7 is 0. Therefore, the total sum mod 7 is (0 + 1 + 3) mod 7 = 4 mod 7. Which matches the previous result! So that's reassuring.

Therefore, regardless of the method, the remainder is 4.

Alternatively, maybe I can check with another approach. Let me compute the sum modulo 7 directly using the formula sum = 100² mod 7.

So 100 mod 7: Let's compute that. 7*14=98, so 100 - 98 = 2, so 100 ≡ 2 mod 7. Therefore, 100² ≡ 2² = 4 mod 7. Which again gives 4.

So this third method also gives 4.

Therefore, all methods confirm that the remainder is 4.

Therefore, I think confidently that the answer is 4.

Just to recap:

Method 1: Sum = n² = 100² = 10,000. 10,000 mod 7 = 4.

Method 2: Sum using arithmetic series formula (n/2)(first + last) = 100/2*(1 + 199) = 50*200 = 10,000. 10,000 mod 7 = 4.

Method 3: Direct modulo computation by breaking down the sum into cycles of residues mod 7, which also gave 4.

Therefore, answer is 4.

I think that's thorough. I don't see any mistakes in my reasoning. So I think the answer is 4.

**Final Answer**
\boxed{4}

To solve the problem, we are asked to find the **modulo 7 remainder** of the sum:

$$
1 + 3 + 5 + 7 + \dots + 195 + 197 + 199
$$

---

### Step 1: Determine the Number of Terms

This is an arithmetic sequence of **odd numbers** starting from 1 and ending at 199, with common difference 2.

We can use the formula for the nth term of an arithmetic sequence:

$$
a_n = a_1 + (n-1)d
$$

Here, $ a_1 = 1 $, $ a_n = 199 $, $ d = 2 $, so:

$$
199 = 1 + (n - 1) \cdot 2 \Rightarrow 198 = 2(n - 1) \Rightarrow n - 1 = 99 \Rightarrow n = 100
$$

So, there are **100 terms** in the sequence.

---

### Step 2: Sum of the Sequence

There is a known formula for the sum of the first $ n $ odd numbers:

$$
\text{Sum} = n^2
$$

So, for $ n = 100 $:

$$
\text{Sum} = 100^2 = 10,000
$$

---

### Step 3: Compute $ 10,000 \mod 7 $

Now, we compute the **modulo 7** remainder of 10,000.

We can simplify by computing $ 10,000 \mod 7 $:

We know:

$$
10 \equiv 3 \mod 7 \Rightarrow 10 \equiv 3 \mod 7
\Rightarrow 10^4 \equiv 3^4 \mod 7
$$

Compute:

$$
3^4 = 81
\Rightarrow 81 \mod 7 = 4
$$

Therefore:

$$
10,000 \equiv 10^4 \equiv 3^4 \equiv 81 \equiv 4 \mod 7
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution:
Instead of adding up the sum and finding the residue, we can find the residue of each number to make computation easier.

Each group of 7 numbers would have the sum of residues $1+3+5+0+2+4+6 \equiv 21 \equiv 0 \pmod7$. Since we only have odd numbers in the sum, every $7$ odd numbers is $14$ integers. Because every group has a residue of $7$, we can ignore them.

There are $\left\lfloor \frac{199}{14}\right\rfloor=14$ sets of $14$ integers, which is equivalent to $7$ odd numbers in our sum. This leaves $197$ and $199$, which have residues $1+3 \equiv \boxed{4} \pmod7$.


Explanation:
Okay, let's see. I need to find f inverse of f inverse of 6. Hmm. Let me recall what inverse functions are. If f is a function, then f inverse takes an output of f and gives back the input. So, if f(a) = b, then f inverse(b) = a. Right?

So, given that f(1) = 2, f(2) = 6, and f(3) = 5. Let me write that down:

f(1) = 2

f(2) = 6

f(3) = 5

So, the inverse function f inverse would satisfy:

f inverse(2) = 1, because f(1) = 2.

Similarly, f inverse(6) = 2, since f(2) = 6.

And f inverse(5) = 3, since f(3) = 5.

So, the problem is asking for f inverse(f inverse(6)). Let's break this down step by step.

First, compute f inverse(6). From above, we know that f inverse(6) is 2. Because f(2) = 6. So, f inverse(6) = 2. Then, we need to compute f inverse of that result, which is f inverse(2). And we already know that f inverse(2) is 1, since f(1) = 2.

Wait, so putting it all together: f inverse(f inverse(6)) = f inverse(2) = 1. Is that right? Let me check again to make sure I didn't mix up anything.

First step: f inverse(6). Since f(2)=6, then yes, f inverse(6) is 2. Then, f inverse of 2 is 1, because f(1)=2. Therefore, f inverse(f inverse(6)) is 1. Seems straightforward.

But let me think again if there's any trick here. Sometimes inverse functions can be confusing if the function isn't one-to-one or something, but the problem says that f inverse exists, so f must be bijective (both injective and surjective) on its domain. But since we are given specific values, maybe we don't need to worry about that. The problem gives us f(1)=2, f(2)=6, f(3)=5. So, assuming that these are the only values we need, and that the function is invertible, so it's one-to-one.

So, since f(2)=6, then f inverse(6)=2. Then f inverse(2) is 1. Therefore, the answer is 1. Hmm. Maybe I should check if there is another way to approach this.

Alternatively, maybe I can think about composing the functions. Let me recall that f inverse(f(x)) = x, provided that x is in the domain of f. Similarly, f(f inverse(y)) = y, provided that y is in the range of f.

So, if I have f inverse(f inverse(6)), maybe I can think of it as applying f inverse twice. Let me see. Let me first compute the inner f inverse(6), which is 2, as before. Then, applying f inverse to that gives 1. So, yes, that seems consistent.

Alternatively, if I try to think about it as a composition, let me consider that applying f inverse twice. But I don't know if that helps. Maybe not. Let me check if there's another perspective.

Alternatively, maybe I can create a table of values for f and f inverse.

For f:

Input | Output

1 | 2

2 | 6

3 | 5

For f inverse:

Input | Output

2 | 1

6 | 2

5 | 3

Therefore, if I want to compute f inverse(f inverse(6)), first compute f inverse(6) which is 2, then compute f inverse(2) which is 1. So, again, same result.

Is there any chance that I could have confused the order? Like, maybe thinking that f inverse(6) is something else? Let me check again. The function f maps 2 to 6, so to get back from 6 to 2, it's f inverse(6) = 2. Then f inverse(2) is 1. So, the answer is 1.

Alternatively, maybe I need to check if the function is invertible. For example, if f was not one-to-one, then f inverse wouldn't exist. But since the problem says that f inverse exists, we can assume that f is one-to-one. Given that f(1)=2, f(2)=6, f(3)=5, there's no overlaps in the outputs. Each input maps to a unique output. So, f is one-to-one here. Therefore, the inverse exists.

Therefore, my conclusion is that f inverse(f inverse(6)) is 1.

Wait, let me think once more. Maybe plugging numbers into the inverse function. Let me define f inverse(y) as the x such that f(x) = y. So, for y=6, find x such that f(x)=6. Looking at the given values, f(2)=6, so x=2. So f inverse(6)=2. Then, for y=2, find x such that f(x)=2. Looking at the given values, f(1)=2, so x=1. Therefore, f inverse(2)=1. Hence, f inverse(f inverse(6)) = f inverse(2) = 1. Therefore, the answer is 1.

I think that's solid. I don't see any flaws in this reasoning. So, I think the answer is 1.

**Final Answer**
\boxed{1}

To solve the problem, we are given a function $ f $ and its inverse $ f^{-1} $, along with the following specific values:

$$
f(1) = 2, \quad f(2) = 6, \quad f(3) = 5
$$

We are asked to compute:

$$
f^{-1}(f^{-1}(6))
$$

---

### Step 1: Understand the Meaning of the Inverse Function

By definition, if $ f(a) = b $, then $ f^{-1}(b) = a $.

This means the inverse function essentially "reverses" the mapping of $ f $: it takes the output of $ f $ and gives back the corresponding input.

---

### Step 2: Compute the Inner Function $ f^{-1}(6) $

We know from the given values that $ f(2) = 6 $. So, the inverse function must satisfy:

$$
f^{-1}(6) = 2
$$

---

### Step 3: Compute the Outer Function $ f^{-1}(f^{-1}(6)) = f^{-1}(2) $

Now we compute $ f^{-1}(2) $. From the given values:

$$
f(1) = 2 \Rightarrow f^{-1}(2) = 1
$$

---

### Final Computation

Putting it all together:

$$
f^{-1}(f^{-1}(6)) = f^{-1}(2) = 1
$$

---

### Final Answer

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, let's see. I need to find f inverse of f inverse of 6. Hmm. Let me recall what inverse functions are. If f is a function, then f inverse takes an output of f and gives back the input. So, if f(a) = b, then f inverse(b) = a. Right?

So, given that f(1) = 2, f(2) = 6, and f(3) = 5. Let me write that down:

f(1) = 2

f(2) = 6

f(3) = 5

So, the inverse function f inverse would satisfy:

f inverse(2) = 1, because f(1) = 2.

Similarly, f inverse(6) = 2, since f(2) = 6.

And f inverse(5) = 3, since f(3) = 5.

So, the problem is asking for f inverse(f inverse(6)). Let's break this down step by step.

First, compute f inverse(6). From above, we know that f inverse(6) is 2. Because f(2) = 6. So, f inverse(6) = 2. Then, we need to compute f inverse of that result, which is f inverse(2). And we already know that f inverse(2) is 1, since f(1) = 2.

Wait, so putting it all together: f inverse(f inverse(6)) = f inverse(2) = 1. Is that right? Let me check again to make sure I didn't mix up anything.

First step: f inverse(6). Since f(2)=6, then yes, f inverse(6) is 2. Then, f inverse of 2 is 1, because f(1)=2. Therefore, f inverse(f inverse(6)) is 1. Seems straightforward.

But let me think again if there's any trick here. Sometimes inverse functions can be confusing if the function isn't one-to-one or something, but the problem says that f inverse exists, so f must be bijective (both injective and surjective) on its domain. But since we are given specific values, maybe we don't need to worry about that. The problem gives us f(1)=2, f(2)=6, f(3)=5. So, assuming that these are the only values we need, and that the function is invertible, so it's one-to-one.

So, since f(2)=6, then f inverse(6)=2. Then f inverse(2) is 1. Therefore, the answer is 1. Hmm. Maybe I should check if there is another way to approach this.

Alternatively, maybe I can think about composing the functions. Let me recall that f inverse(f(x)) = x, provided that x is in the domain of f. Similarly, f(f inverse(y)) = y, provided that y is in the range of f.

So, if I have f inverse(f inverse(6)), maybe I can think of it as applying f inverse twice. Let me see. Let me first compute the inner f inverse(6), which is 2, as before. Then, applying f inverse to that gives 1. So, yes, that seems consistent.

Alternatively, if I try to think about it as a composition, let me consider that applying f inverse twice. But I don't know if that helps. Maybe not. Let me check if there's another perspective.

Alternatively, maybe I can create a table of values for f and f inverse.

For f:

Input | Output

1 | 2

2 | 6

3 | 5

For f inverse:

Input | Output

2 | 1

6 | 2

5 | 3

Therefore, if I want to compute f inverse(f inverse(6)), first compute f inverse(6) which is 2, then compute f inverse(2) which is 1. So, again, same result.

Is there any chance that I could have confused the order? Like, maybe thinking that f inverse(6) is something else? Let me check again. The function f maps 2 to 6, so to get back from 6 to 2, it's f inverse(6) = 2. Then f inverse(2) is 1. So, the answer is 1.

Alternatively, maybe I need to check if the function is invertible. For example, if f was not one-to-one, then f inverse wouldn't exist. But since the problem says that f inverse exists, we can assume that f is one-to-one. Given that f(1)=2, f(2)=6, f(3)=5, there's no overlaps in the outputs. Each input maps to a unique output. So, f is one-to-one here. Therefore, the inverse exists.

Therefore, my conclusion is that f inverse(f inverse(6)) is 1.

Wait, let me think once more. Maybe plugging numbers into the inverse function. Let me define f inverse(y) as the x such that f(x) = y. So, for y=6, find x such that f(x)=6. Looking at the given values, f(2)=6, so x=2. So f inverse(6)=2. Then, for y=2, find x such that f(x)=2. Looking at the given values, f(1)=2, so x=1. Therefore, f inverse(2)=1. Hence, f inverse(f inverse(6)) = f inverse(2) = 1. Therefore, the answer is 1.

I think that's solid. I don't see any flaws in this reasoning. So, I think the answer is 1.

**Final Answer**
\boxed{1}

To solve the problem, we are given a function $ f $ and its inverse $ f^{-1} $, along with the following specific values:

$$
f(1) = 2, \quad f(2) = 6, \quad f(3) = 5
$$

We are asked to compute:

$$
f^{-1}(f^{-1}(6))
$$

---

### Step 1: Understand the Meaning of the Inverse Function

By definition, if $ f(a) = b $, then $ f^{-1}(b) = a $.

This means the inverse function essentially "reverses" the mapping of $ f $: it takes the output of $ f $ and gives back the corresponding input.

---

### Step 2: Compute the Inner Function $ f^{-1}(6) $

We know from the given values that $ f(2) = 6 $. So, the inverse function must satisfy:

$$
f^{-1}(6) = 2
$$

---

### Step 3: Compute the Outer Function $ f^{-1}(f^{-1}(6)) = f^{-1}(2) $

Now we compute $ f^{-1}(2) $. From the given values:

$$
f(1) = 2 \Rightarrow f^{-1}(2) = 1
$$

---

### Final Computation

Putting it all together:

$$
f^{-1}(f^{-1}(6)) = f^{-1}(2) = 1
$$

---

### Final Answer

$$
\boxed{1}
$$

Solution: Since $f(2) = 6$, we have $f^{-1}(6)=2$. (Note that the hypothesis that $f$ has an inverse implies that there are no other values of $x$ with $f(x) = 6$.)  Similarly, $f(1) =2$ implies $f^{-1}(2)=1$.   So $f^{-1}(f^{-1}(6))=f^{-1}(2)=\boxed{1}$.