Explanation:
Okay, so I need to find the value of $ xy + yz + zx $ given these three equations:

1. $ xyz = 4 $
2. $ x^3 + y^3 + z^3 = 4 $
3. $ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z = 12 $

Wait, hold on, the third equation seems a bit confusing. Let me check again. The user wrote:

"xy² + x²y + xz² + x²z + yz² + yz² + y²z = 12"

Wait, that seems like there's a typo. Let me count the terms. The original third equation is:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = 12 $

Wait, actually, the user wrote "yz² + yz² + y²z" which would be two yz² terms? That seems like a mistake. Let me check again.

Original problem:

"Suppose that $x,$ $y,$ and $z$ satisfy the equations

$$
\begin{align*}
xyz &= 4, \\
x^3 + y^3 + z^3 &= 4, \\
xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z &= 12.
\end{align*}
$$

Calculate the value of $xy + yz + zx$."

Wait, so the third equation is written as:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z = 12 $

But that would be:

xy² + x²y + xz² + x²z + yz² + yz² + y²z. So that's two yz² terms? That seems like a mistake. Maybe it's supposed to be:

xy² + x²y + xz² + x²z + yz² + y²z = 12?

Which would be symmetric. Let me check the original problem again. The user wrote:

"xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z = 12"

Wait, so the third term is xz², then x²z, then yz², then yz² again, then y²z. So that's two yz² terms? That seems like a typo. Maybe the user meant:

xy² + x²y + xz² + x²z + yz² + y²z = 12?

Which is symmetric. Let me check the number of terms. The original third equation is supposed to be symmetric? Let me think.

Alternatively, maybe the user made a typo and the third equation is supposed to be:

xy² + x²y + xz² + x²z + yz² + y²z = 12?

Which is symmetric. Let me check the number of terms. Let me count:

xy², x²y, xz², x²z, yz², y²z. That's six terms. So that's symmetric. So maybe the original problem had a typo with an extra yz²? Maybe the user meant that. Alternatively, maybe the original problem is correct, but with two yz² terms. Let me check.

But if there are two yz² terms, then the equation would be:

xy² + x²y + xz² + x²z + 2yz² + y²z = 12. But that seems asymmetric. Alternatively, maybe the user meant:

xy² + x²y + xz² + x²z + yz² + y²z = 12. Which is symmetric. Let me assume that. Because otherwise, the problem is asymmetric and maybe harder. Let me check the original problem again.

Wait, the original problem is written as:

"xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z = 12"

So, the user wrote "yz^2 + yz^2 + y^2 z". So that's two yz² terms and one y²z. So that's three terms? Wait, no. Let me parse the original third equation again:

Original third equation:

xy² + x²y + xz² + x²z + yz² + yz² + y²z = 12

So, breaking it down:

First term: xy²

Second term: x²y

Third term: xz²

Fourth term: x²z

Fifth term: yz²

Sixth term: yz²

Seventh term: y²z

So, total of seven terms? Wait, but that seems odd. Let me check again. Wait, maybe the user made a typo in the original problem. Maybe the third equation is supposed to be:

xy² + x²y + xz² + x²z + yz² + y²z = 12

Which is six terms. So, maybe the user had a typo and wrote an extra yz². Alternatively, maybe the original problem is correct. Let me think.

Alternatively, maybe the third equation is:

xy² + x²y + xz² + x²z + yz² + y²z = 12

Which is symmetric. Let me check if that's possible. Let me assume that. Because otherwise, the problem is asymmetric and maybe harder. Let me proceed with that assumption. If I get stuck, I can check again.

So, assuming the third equation is:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = 12 $

So, that's symmetric. Let me note that.

So, given that, we need to find $ xy + yz + zx $.

So, given that, let me recall some algebraic identities.

First, we know that:

$ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) $

Given that $ x^3 + y^3 + z^3 = 4 $ and $ xyz = 4 $, so substituting into the identity:

$ 4 - 3*4 = (x + y + z)(x^2 + y^2 + z^2 - (xy + yz + zx)) $

Calculating left side: 4 - 12 = -8

So:

$ -8 = (x + y + z)(x^2 + y^2 + z^2 - (xy + yz + zx)) $

Let me denote $ S = x + y + z $, $ P = xy + yz + zx $, and $ Q = xyz = 4 $. Also, $ x^2 + y^2 + z^2 = S^2 - 2P $.

Therefore, substituting into the equation:

$ -8 = S( (S^2 - 2P) - P ) = S(S^2 - 3P) $

So:

$ -8 = S^3 - 3SP $

So, equation (A): $ S^3 - 3SP = -8 $

Now, moving to the third equation. Let me analyze the third equation:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = 12 $

Let me note that this expression can be written as:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = (xy^2 + x^2 y) + (xz^2 + x^2 z) + (yz^2 + y^2 z) $

Each pair like $ xy^2 + x^2 y = xy(y + x) $, similarly for others. So:

= $ xy(x + y) + xz(x + z) + yz(y + z) $

Alternatively, maybe there's another way to express this. Let me think.

Alternatively, note that:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = (x + y + z)(xy + yz + zx) - 3xyz $

Wait, let me check that. Let me compute $ (x + y + z)(xy + yz + zx) $:

= $ x(xy) + x(yz) + x(zx) + y(xy) + y(yz) + y(zx) + z(xy) + z(yz) + z(zx) $

Wait, that's:

= $ x^2 y + xyz + x^2 z + xy^2 + y^2 z + yzx + zxy + y z^2 + z^2 x $

Simplify:

= $ x^2 y + x^2 z + xy^2 + y^2 z + xz^2 + y z^2 + 3xyz $

Therefore, $ (x + y + z)(xy + yz + zx) = x^2 y + x^2 z + xy^2 + y^2 z + xz^2 + y z^2 + 3xyz $

Therefore, the third equation is:

$ x^2 y + x^2 z + xy^2 + y^2 z + xz^2 + y z^2 = 12 $

Which is equal to $ (x + y + z)(xy + yz + zx) - 3xyz $

Therefore, substituting:

$ (S)(P) - 3Q = 12 $

Given that Q = 4, so:

$ SP - 12 = 12 $

Therefore:

$ SP = 24 $

So, equation (B): $ SP = 24 $

So now, from equation (A): $ S^3 - 3SP = -8 $

But since SP = 24, substitute into equation (A):

$ S^3 - 3*24 = -8 $

So:

$ S^3 - 72 = -8 $

Therefore:

$ S^3 = 64 $

Therefore:

$ S = \sqrt[3]{64} = 4 $

So, S = 4.

Then, from equation (B): SP = 24 => 4 * P = 24 => P = 6.

Therefore, the value of $ xy + yz + zx = P = 6 $.

Wait, so is that the answer? Let me check again.

Let me recap:

We had the third equation, which we transformed into SP - 3Q = 12. Since Q = 4, SP = 24. Then, using the identity for x³ + y³ + z³, we found S³ - 3SP = -8, which with SP = 24 gives S³ = 64, so S = 4, then P = 6. Therefore, the answer is 6. That seems straightforward. But let me check if my transformation of the third equation was correct.

Original third equation:

xy² + x²y + xz² + x²z + yz² + y²z = 12

Which we said equals (x + y + z)(xy + yz + zx) - 3xyz. Let me verify with numbers. Let me take simple numbers where I know the values.

Suppose x = y = z = 1. Then:

Left side: 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 = 6

Right side: (1 + 1 + 1)(1*1 + 1*1 + 1*1) - 3*1*1*1 = 3*3 - 3 = 9 - 3 = 6. Correct.

Another example: x=2, y=1, z=0. Then left side:

xy² + x²y + xz² + x²z + yz² + y²z = 2*1 + 4*1 + 0 + 0 + 0 + 1*0 = 2 + 4 = 6

Right side: (2 + 1 + 0)(2*1 + 1*0 + 0*2) - 3*2*1*0 = 3*(2 + 0 + 0) - 0 = 6. Correct. So the identity holds. Therefore, my transformation is correct.

Therefore, the answer is 6. Therefore, the value of $ xy + yz + zx $ is 6. Therefore, the answer is \boxed{6}.

But let me check if there are any other possibilities. For example, when we solved S³ = 64, S could be 4, but in complex numbers, there are other roots. However, since we are dealing with real numbers (assuming the problem is in real numbers), S = 4 is the only real solution. Therefore, P = 6 is the answer.

Alternatively, let me check if the original third equation was different. Suppose that the original third equation had an extra term. For example, if there were two yz² terms, then the equation would be different. But since the problem as stated by the user has that typo, but assuming that it's a typo and the correct equation is symmetric, then the answer is 6. However, if the original equation was different, the answer might be different. But given that the problem is from a competition or textbook, it's more likely that the third equation is symmetric, so the answer is 6.

Alternatively, let me check if with S = 4, P = 6, and Q = 4, whether there exist real numbers x, y, z satisfying the equations.

Let me try to find such numbers. Let me assume that x, y, z are roots of the cubic equation t³ - St² + Pt - Q = 0, so t³ - 4t² + 6t - 4 = 0.

Let me try to factor this cubic. Let me check for rational roots. By Rational Root Theorem, possible roots are ±1, ±2, ±4.

Testing t=1: 1 - 4 + 6 - 4 = -1 ≠ 0

t=2: 8 - 16 + 12 - 4 = 0. Yes! t=2 is a root.

Therefore, we can factor as (t - 2)(t² - 2t + 2) = 0.

Therefore, the roots are t=2 and roots of t² - 2t + 2 = 0, which are [2 ± sqrt(4 - 8)] / 2 = 1 ± i. So complex numbers.

Therefore, if we take x=2, y=1+i, z=1-i, then check the equations.

First, xyz = 2*(1+i)*(1-i) = 2*(1 +1) = 4. Correct.

x³ + y³ + z³: Let's compute.

x³ = 8.

y³ = (1 + i)^3 = 1 + 3i + 3i² + i³ = 1 + 3i - 3 - i = (-2 + 2i)

z³ = (1 - i)^3 = 1 - 3i + 3i² - i³ = 1 - 3i - 3 + i = (-2 - 2i)

Therefore, x³ + y³ + z³ = 8 + (-2 + 2i) + (-2 - 2i) = 8 - 4 = 4. Correct.

Third equation: Let's compute xy² + x²y + xz² + x²z + yz² + y²z.

First, compute each term:

xy²: x=2, y=1+i. y² = (1+i)^2 = 2i. So 2*(2i) = 4i

x²y: x²=4, y=1+i. 4*(1+i) = 4 + 4i

xz²: z=1 - i. z² = (1 - i)^2 = -2i. So 2*(-2i) = -4i

x²z: 4*(1 - i) = 4 - 4i

yz²: y=1+i, z²=-2i. So (1+i)*(-2i) = -2i - 2i² = -2i + 2 = 2 - 2i

y²z: y²=2i, z=1 - i. 2i*(1 - i) = 2i - 2i² = 2i + 2 = 2 + 2i

Now, adding all terms:

4i + (4 + 4i) + (-4i) + (4 - 4i) + (2 - 2i) + (2 + 2i)

Let me compute term by term:

Start with 4i.

Add 4 + 4i: total is 4 + 8i

Add -4i: 4 + 4i

Add 4 - 4i: 8 + 0i

Add 2 - 2i: 10 - 2i

Add 2 + 2i: 12 + 0i

So total is 12. Which matches the third equation. Therefore, even though the numbers are complex, they satisfy the equations. Therefore, the value of P = 6 is correct.

But the problem didn't specify that x, y, z are real numbers. However, even if they are complex, the value of $ xy + yz + zx $ is still 6. Therefore, regardless of the nature of x, y, z, the answer is 6.

Therefore, the answer is \boxed{6}.

**Final Answer**
\boxed{6}

To solve the problem, we are given the following system of equations:

$$
\begin{align*}
(1) \quad & xyz = 4, \\
(2) \quad & x^3 + y^3 + z^3 = 4, \\
(3) \quad & xy^2 + x^2y + xz^2 + x^2z + yz^2 + y^2z = 12.
\end{align*}
$$

We are asked to find the value of $ xy + yz + zx $.

---

### Step 1: Use the identity for sum of cubes

Recall the identity:

$$
x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)
$$

Let us denote:

- $ S = x + y + z $
- $ P = xy + yz + zx $
- $ Q = xyz = 4 $

Then, from the identity:

$$
x^3 + y^3 + z^3 - 3xyz = S(x^2 + y^2 + z^2 - P)
$$

We also know that:

$$
x^2 + y^2 + z^2 = S^2 - 2P
$$

So the identity becomes:

$$
x^3 + y^3 + z^3 - 3Q = S(S^2 - 3P)
$$

Substitute the known values:

$$
4 - 3 \cdot 4 = S(S^2 - 3P) \Rightarrow -8 = S^3 - 3SP \tag{1}
$$

---

### Step 2: Analyze the third equation

We are given:

$$
xy^2 + x^2y + xz^2 + x^2z + yz^2 + y^2z = 12
$$

This expression can be rewritten using the identity:

$$
(x + y + z)(xy + yz + zx) = x^2y + x^2z + xy^2 + y^2z + xz^2 + yz^2 + 3xyz
$$

So:

$$
xy^2 + x^2y + xz^2 + x^2z + yz^2 + y^2z = (x + y + z)(xy + yz + zx) - 3xyz
$$

Substitute known values:

$$
SP - 3Q = 12 \Rightarrow SP - 12 = 12 \Rightarrow SP = 24 \tag{2}
$$

---

### Step 3: Solve the system of equations

From (1):
$$
S^3 - 3SP = -8
$$

From (2):
$$
SP = 24
$$

Substitute into (1):

$$
S^3 - 3 \cdot 24 = -8 \Rightarrow S^3 = 64 \Rightarrow S = 4
$$

Now use (2) to find $ P $:

$$
SP = 24 \Rightarrow 4P = 24 \Rightarrow P = 6
$$

---

### Final Answer

$$
\boxed{6}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of $ xy + yz + zx $ given these three equations:

1. $ xyz = 4 $
2. $ x^3 + y^3 + z^3 = 4 $
3. $ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z = 12 $

Wait, hold on, the third equation seems a bit confusing. Let me check again. The user wrote:

"xy² + x²y + xz² + x²z + yz² + yz² + y²z = 12"

Wait, that seems like there's a typo. Let me count the terms. The original third equation is:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = 12 $

Wait, actually, the user wrote "yz² + yz² + y²z" which would be two yz² terms? That seems like a mistake. Let me check again.

Original problem:

"Suppose that $x,$ $y,$ and $z$ satisfy the equations

$$
\begin{align*}
xyz &= 4, \\
x^3 + y^3 + z^3 &= 4, \\
xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z &= 12.
\end{align*}
$$

Calculate the value of $xy + yz + zx$."

Wait, so the third equation is written as:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z = 12 $

But that would be:

xy² + x²y + xz² + x²z + yz² + yz² + y²z. So that's two yz² terms? That seems like a mistake. Maybe it's supposed to be:

xy² + x²y + xz² + x²z + yz² + y²z = 12?

Which would be symmetric. Let me check the original problem again. The user wrote:

"xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z = 12"

Wait, so the third term is xz², then x²z, then yz², then yz² again, then y²z. So that's two yz² terms? That seems like a typo. Maybe the user meant:

xy² + x²y + xz² + x²z + yz² + y²z = 12?

Which is symmetric. Let me check the number of terms. The original third equation is supposed to be symmetric? Let me think.

Alternatively, maybe the user made a typo and the third equation is supposed to be:

xy² + x²y + xz² + x²z + yz² + y²z = 12?

Which is symmetric. Let me check the number of terms. Let me count:

xy², x²y, xz², x²z, yz², y²z. That's six terms. So that's symmetric. So maybe the original problem had a typo with an extra yz²? Maybe the user meant that. Alternatively, maybe the original problem is correct, but with two yz² terms. Let me check.

But if there are two yz² terms, then the equation would be:

xy² + x²y + xz² + x²z + 2yz² + y²z = 12. But that seems asymmetric. Alternatively, maybe the user meant:

xy² + x²y + xz² + x²z + yz² + y²z = 12. Which is symmetric. Let me assume that. Because otherwise, the problem is asymmetric and maybe harder. Let me check the original problem again.

Wait, the original problem is written as:

"xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + yz^2 + y^2 z = 12"

So, the user wrote "yz^2 + yz^2 + y^2 z". So that's two yz² terms and one y²z. So that's three terms? Wait, no. Let me parse the original third equation again:

Original third equation:

xy² + x²y + xz² + x²z + yz² + yz² + y²z = 12

So, breaking it down:

First term: xy²

Second term: x²y

Third term: xz²

Fourth term: x²z

Fifth term: yz²

Sixth term: yz²

Seventh term: y²z

So, total of seven terms? Wait, but that seems odd. Let me check again. Wait, maybe the user made a typo in the original problem. Maybe the third equation is supposed to be:

xy² + x²y + xz² + x²z + yz² + y²z = 12

Which is six terms. So, maybe the user had a typo and wrote an extra yz². Alternatively, maybe the original problem is correct. Let me think.

Alternatively, maybe the third equation is:

xy² + x²y + xz² + x²z + yz² + y²z = 12

Which is symmetric. Let me check if that's possible. Let me assume that. Because otherwise, the problem is asymmetric and maybe harder. Let me proceed with that assumption. If I get stuck, I can check again.

So, assuming the third equation is:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = 12 $

So, that's symmetric. Let me note that.

So, given that, we need to find $ xy + yz + zx $.

So, given that, let me recall some algebraic identities.

First, we know that:

$ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) $

Given that $ x^3 + y^3 + z^3 = 4 $ and $ xyz = 4 $, so substituting into the identity:

$ 4 - 3*4 = (x + y + z)(x^2 + y^2 + z^2 - (xy + yz + zx)) $

Calculating left side: 4 - 12 = -8

So:

$ -8 = (x + y + z)(x^2 + y^2 + z^2 - (xy + yz + zx)) $

Let me denote $ S = x + y + z $, $ P = xy + yz + zx $, and $ Q = xyz = 4 $. Also, $ x^2 + y^2 + z^2 = S^2 - 2P $.

Therefore, substituting into the equation:

$ -8 = S( (S^2 - 2P) - P ) = S(S^2 - 3P) $

So:

$ -8 = S^3 - 3SP $

So, equation (A): $ S^3 - 3SP = -8 $

Now, moving to the third equation. Let me analyze the third equation:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = 12 $

Let me note that this expression can be written as:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = (xy^2 + x^2 y) + (xz^2 + x^2 z) + (yz^2 + y^2 z) $

Each pair like $ xy^2 + x^2 y = xy(y + x) $, similarly for others. So:

= $ xy(x + y) + xz(x + z) + yz(y + z) $

Alternatively, maybe there's another way to express this. Let me think.

Alternatively, note that:

$ xy^2 + x^2 y + xz^2 + x^2 z + yz^2 + y^2 z = (x + y + z)(xy + yz + zx) - 3xyz $

Wait, let me check that. Let me compute $ (x + y + z)(xy + yz + zx) $:

= $ x(xy) + x(yz) + x(zx) + y(xy) + y(yz) + y(zx) + z(xy) + z(yz) + z(zx) $

Wait, that's:

= $ x^2 y + xyz + x^2 z + xy^2 + y^2 z + yzx + zxy + y z^2 + z^2 x $

Simplify:

= $ x^2 y + x^2 z + xy^2 + y^2 z + xz^2 + y z^2 + 3xyz $

Therefore, $ (x + y + z)(xy + yz + zx) = x^2 y + x^2 z + xy^2 + y^2 z + xz^2 + y z^2 + 3xyz $

Therefore, the third equation is:

$ x^2 y + x^2 z + xy^2 + y^2 z + xz^2 + y z^2 = 12 $

Which is equal to $ (x + y + z)(xy + yz + zx) - 3xyz $

Therefore, substituting:

$ (S)(P) - 3Q = 12 $

Given that Q = 4, so:

$ SP - 12 = 12 $

Therefore:

$ SP = 24 $

So, equation (B): $ SP = 24 $

So now, from equation (A): $ S^3 - 3SP = -8 $

But since SP = 24, substitute into equation (A):

$ S^3 - 3*24 = -8 $

So:

$ S^3 - 72 = -8 $

Therefore:

$ S^3 = 64 $

Therefore:

$ S = \sqrt[3]{64} = 4 $

So, S = 4.

Then, from equation (B): SP = 24 => 4 * P = 24 => P = 6.

Therefore, the value of $ xy + yz + zx = P = 6 $.

Wait, so is that the answer? Let me check again.

Let me recap:

We had the third equation, which we transformed into SP - 3Q = 12. Since Q = 4, SP = 24. Then, using the identity for x³ + y³ + z³, we found S³ - 3SP = -8, which with SP = 24 gives S³ = 64, so S = 4, then P = 6. Therefore, the answer is 6. That seems straightforward. But let me check if my transformation of the third equation was correct.

Original third equation:

xy² + x²y + xz² + x²z + yz² + y²z = 12

Which we said equals (x + y + z)(xy + yz + zx) - 3xyz. Let me verify with numbers. Let me take simple numbers where I know the values.

Suppose x = y = z = 1. Then:

Left side: 1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1 = 6

Right side: (1 + 1 + 1)(1*1 + 1*1 + 1*1) - 3*1*1*1 = 3*3 - 3 = 9 - 3 = 6. Correct.

Another example: x=2, y=1, z=0. Then left side:

xy² + x²y + xz² + x²z + yz² + y²z = 2*1 + 4*1 + 0 + 0 + 0 + 1*0 = 2 + 4 = 6

Right side: (2 + 1 + 0)(2*1 + 1*0 + 0*2) - 3*2*1*0 = 3*(2 + 0 + 0) - 0 = 6. Correct. So the identity holds. Therefore, my transformation is correct.

Therefore, the answer is 6. Therefore, the value of $ xy + yz + zx $ is 6. Therefore, the answer is \boxed{6}.

But let me check if there are any other possibilities. For example, when we solved S³ = 64, S could be 4, but in complex numbers, there are other roots. However, since we are dealing with real numbers (assuming the problem is in real numbers), S = 4 is the only real solution. Therefore, P = 6 is the answer.

Alternatively, let me check if the original third equation was different. Suppose that the original third equation had an extra term. For example, if there were two yz² terms, then the equation would be different. But since the problem as stated by the user has that typo, but assuming that it's a typo and the correct equation is symmetric, then the answer is 6. However, if the original equation was different, the answer might be different. But given that the problem is from a competition or textbook, it's more likely that the third equation is symmetric, so the answer is 6.

Alternatively, let me check if with S = 4, P = 6, and Q = 4, whether there exist real numbers x, y, z satisfying the equations.

Let me try to find such numbers. Let me assume that x, y, z are roots of the cubic equation t³ - St² + Pt - Q = 0, so t³ - 4t² + 6t - 4 = 0.

Let me try to factor this cubic. Let me check for rational roots. By Rational Root Theorem, possible roots are ±1, ±2, ±4.

Testing t=1: 1 - 4 + 6 - 4 = -1 ≠ 0

t=2: 8 - 16 + 12 - 4 = 0. Yes! t=2 is a root.

Therefore, we can factor as (t - 2)(t² - 2t + 2) = 0.

Therefore, the roots are t=2 and roots of t² - 2t + 2 = 0, which are [2 ± sqrt(4 - 8)] / 2 = 1 ± i. So complex numbers.

Therefore, if we take x=2, y=1+i, z=1-i, then check the equations.

First, xyz = 2*(1+i)*(1-i) = 2*(1 +1) = 4. Correct.

x³ + y³ + z³: Let's compute.

x³ = 8.

y³ = (1 + i)^3 = 1 + 3i + 3i² + i³ = 1 + 3i - 3 - i = (-2 + 2i)

z³ = (1 - i)^3 = 1 - 3i + 3i² - i³ = 1 - 3i - 3 + i = (-2 - 2i)

Therefore, x³ + y³ + z³ = 8 + (-2 + 2i) + (-2 - 2i) = 8 - 4 = 4. Correct.

Third equation: Let's compute xy² + x²y + xz² + x²z + yz² + y²z.

First, compute each term:

xy²: x=2, y=1+i. y² = (1+i)^2 = 2i. So 2*(2i) = 4i

x²y: x²=4, y=1+i. 4*(1+i) = 4 + 4i

xz²: z=1 - i. z² = (1 - i)^2 = -2i. So 2*(-2i) = -4i

x²z: 4*(1 - i) = 4 - 4i

yz²: y=1+i, z²=-2i. So (1+i)*(-2i) = -2i - 2i² = -2i + 2 = 2 - 2i

y²z: y²=2i, z=1 - i. 2i*(1 - i) = 2i - 2i² = 2i + 2 = 2 + 2i

Now, adding all terms:

4i + (4 + 4i) + (-4i) + (4 - 4i) + (2 - 2i) + (2 + 2i)

Let me compute term by term:

Start with 4i.

Add 4 + 4i: total is 4 + 8i

Add -4i: 4 + 4i

Add 4 - 4i: 8 + 0i

Add 2 - 2i: 10 - 2i

Add 2 + 2i: 12 + 0i

So total is 12. Which matches the third equation. Therefore, even though the numbers are complex, they satisfy the equations. Therefore, the value of P = 6 is correct.

But the problem didn't specify that x, y, z are real numbers. However, even if they are complex, the value of $ xy + yz + zx $ is still 6. Therefore, regardless of the nature of x, y, z, the answer is 6.

Therefore, the answer is \boxed{6}.

**Final Answer**
\boxed{6}

To solve the problem, we are given the following system of equations:

$$
\begin{align*}
(1) \quad & xyz = 4, \\
(2) \quad & x^3 + y^3 + z^3 = 4, \\
(3) \quad & xy^2 + x^2y + xz^2 + x^2z + yz^2 + y^2z = 12.
\end{align*}
$$

We are asked to find the value of $ xy + yz + zx $.

---

### Step 1: Use the identity for sum of cubes

Recall the identity:

$$
x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)
$$

Let us denote:

- $ S = x + y + z $
- $ P = xy + yz + zx $
- $ Q = xyz = 4 $

Then, from the identity:

$$
x^3 + y^3 + z^3 - 3xyz = S(x^2 + y^2 + z^2 - P)
$$

We also know that:

$$
x^2 + y^2 + z^2 = S^2 - 2P
$$

So the identity becomes:

$$
x^3 + y^3 + z^3 - 3Q = S(S^2 - 3P)
$$

Substitute the known values:

$$
4 - 3 \cdot 4 = S(S^2 - 3P) \Rightarrow -8 = S^3 - 3SP \tag{1}
$$

---

### Step 2: Analyze the third equation

We are given:

$$
xy^2 + x^2y + xz^2 + x^2z + yz^2 + y^2z = 12
$$

This expression can be rewritten using the identity:

$$
(x + y + z)(xy + yz + zx) = x^2y + x^2z + xy^2 + y^2z + xz^2 + yz^2 + 3xyz
$$

So:

$$
xy^2 + x^2y + xz^2 + x^2z + yz^2 + y^2z = (x + y + z)(xy + yz + zx) - 3xyz
$$

Substitute known values:

$$
SP - 3Q = 12 \Rightarrow SP - 12 = 12 \Rightarrow SP = 24 \tag{2}
$$

---

### Step 3: Solve the system of equations

From (1):
$$
S^3 - 3SP = -8
$$

From (2):
$$
SP = 24
$$

Substitute into (1):

$$
S^3 - 3 \cdot 24 = -8 \Rightarrow S^3 = 64 \Rightarrow S = 4
$$

Now use (2) to find $ P $:

$$
SP = 24 \Rightarrow 4P = 24 \Rightarrow P = 6
$$

---

### Final Answer

$$
\boxed{6}
$$

Solution:
Let $s_1 = x + y + z$ and $s_2 = xy + xz + yz.$  Then
\begin{align*}
s_1 s_2 &= (x + y + z)(xy + xz + yz) \\
&= x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2 + 3xyz \\
&= 12 + 3 \cdot 4 = 24.
\end{align*}Also,
\begin{align*}
s_1^3 &= (x + y + z)^3 \\
&= (x^3 + y^3 + z^3) + 3(x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2) + 6xyz \\
&= 4 + 3 \cdot 12 + 6 \cdot 4 = 64,
\end{align*}so $s_1 = 4.$  Hence, $s_2 = \frac{24}{s_1} = \boxed{6}.$


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, 4p = 8, so p = 2. That means the focus is at (0, 2), but maybe that's not directly relevant here.

Now, the problem says an equilateral triangle is inscribed in the parabola, with one vertex at the vertex of the parabola, which is (0,0). Let me denote the three vertices of the triangle as A, B, and C. Let's say point A is at (0,0). Then points B and C are somewhere on the parabola.

Since it's an equilateral triangle, all sides must be equal. So AB = BC = CA. But since A is at (0,0), I need to find coordinates for B and C on the parabola such that the distances AB, BC, and CA are equal.

Let me denote coordinates for points B and C. Let me assume that points B and C are symmetric with respect to the y-axis. Because the parabola is symmetric about the y-axis, and if we have an equilateral triangle with one vertex at the origin, maybe the other two vertices are symmetric. That might simplify things.

So let me suppose that point B is (a, b) and point C is (-a, b), since they are symmetric across the y-axis. Then, since they lie on the parabola $ x^2 = 8y $, their coordinates must satisfy this equation. Therefore, for point B: $ a^2 = 8b $, so $ b = a^2 / 8 $. Similarly for point C, same thing.

Now, let me compute the distances between these points. First, AB and AC. Since A is (0,0), the distance AB is sqrt[(a - 0)^2 + (b - 0)^2] = sqrt(a² + b²). Similarly, AC is the same because of symmetry.

Then, the distance BC is between (a, b) and (-a, b). That distance is sqrt[(-a - a)^2 + (b - b)^2] = sqrt[(-2a)^2 + 0] = sqrt[4a²] = 2|a|. Since a is a coordinate on the parabola, it can be positive or negative, but since we squared it, we can assume a is positive without loss of generality.

So, since the triangle is equilateral, AB = BC. Therefore:

sqrt(a² + b²) = 2a.

But we also know that b = a² / 8. Let me substitute that into the equation:

sqrt(a² + (a² / 8)^2) = 2a.

Let me square both sides to eliminate the square root:

a² + (a^4) / 64 = 4a².

Simplify:

(a^4)/64 + a² - 4a² = 0

Which becomes:

(a^4)/64 - 3a² = 0

Multiply both sides by 64 to eliminate the denominator:

a^4 - 192a² = 0

Factor:

a²(a² - 192) = 0

So, solutions are a² = 0 or a² = 192. Since a = 0 would give points B and C both at (0,0), which would collapse the triangle, we discard a² = 0. Therefore, a² = 192, so a = sqrt(192) or -sqrt(192). Since we assumed a positive, a = sqrt(192).

Simplify sqrt(192): sqrt(64*3) = 8*sqrt(3). Therefore, a = 8√3. Then, b = a² / 8 = (192)/8 = 24. Therefore, points B and C are (8√3, 24) and (-8√3, 24).

Now, let me check if the distance AB equals BC. Let me compute AB:

sqrt[(8√3)^2 + (24)^2] = sqrt(64*3 + 576) = sqrt(192 + 576) = sqrt(768). Simplify sqrt(768): sqrt(256*3) = 16√3.

Distance BC is 2a = 2*(8√3) = 16√3. So yes, AB = BC. But wait, the triangle is supposed to be equilateral, so we also need to check that AC = AB and BC = AC, which they are by symmetry. But also, we need to check that the third side, which is AC, is equal. Wait, AC is same as AB, so that's already covered. However, we also need to check that the distance between B and C is equal to AB. Which we did, and it is. So that seems okay.

But wait, hold on. Wait, in an equilateral triangle, all sides are equal. So AB = BC = CA. But here, we have AB = BC, and CA is same as AB. So that's okay. But wait, but maybe there's a mistake here? Because if points B and C are symmetric across the y-axis, then the triangle is isoceles with AB = AC, but for it to be equilateral, BC must also equal AB. Which we have done. So according to this, the side length is 16√3. But let me check if that's correct.

Wait, but let me think again. If the triangle is equilateral, then all angles are 60 degrees. However, in this case, with points at (0,0), (8√3, 24), (-8√3, 24), is the angle at A 60 degrees? Let me check.

Alternatively, maybe there's a mistake in assuming that points B and C are symmetric across the y-axis. Maybe that's not the case? Wait, but given the symmetry of the parabola, maybe that's the only way? Or maybe there's another configuration?

Wait, but let me think again. Suppose that the triangle is not symmetric. Maybe the other two vertices are not symmetric? But then how would we find them? It might be more complicated. However, given that the parabola is symmetric about the y-axis, and the triangle has one vertex at the origin, it's plausible that the other two vertices are symmetric. But maybe there's another triangle where they are not symmetric? But since the problem says "an equilateral triangle", maybe there are multiple such triangles, but the question is to find the side length. However, maybe the symmetric case is the only one? Or maybe not?

Wait, but let me check if the triangle I found is actually equilateral. Let me compute all sides.

AB: distance from (0,0) to (8√3, 24): sqrt[(8√3)^2 + 24^2] = sqrt(192 + 576) = sqrt(768) = 16√3.

BC: distance between (8√3, 24) and (-8√3, 24): sqrt[( -8√3 - 8√3)^2 + (24 - 24)^2] = sqrt[(-16√3)^2 + 0] = sqrt(768) = 16√3.

AC: same as AB, so 16√3. So yes, all sides are equal. Therefore, this is an equilateral triangle. Therefore, the side length is 16√3. But wait, let me check if this is correct. However, I have a feeling that maybe there's a mistake here because I remember that in some cases, when dealing with parabolas and equilateral triangles, there might be another configuration. Let me think again.

Alternatively, maybe I need to consider that the triangle is not symmetric? Let me try to approach this more generally.

Let me suppose that the three vertices are A(0,0), B(x1, y1), and C(x2, y2), all lying on the parabola x² = 8y. Therefore, y1 = x1² / 8 and y2 = x2² / 8.

Since the triangle is equilateral, the distances AB, BC, and CA must be equal. So:

AB² = (x1 - 0)^2 + (y1 - 0)^2 = x1² + y1² = x1² + (x1² / 8)^2 = x1² + x1^4 / 64.

Similarly, AC² = x2² + (x2² / 8)^2.

And BC² = (x2 - x1)^2 + (y2 - y1)^2 = (x2 - x1)^2 + (x2² / 8 - x1² / 8)^2.

Since AB = BC = CA, so AB² = BC² and AB² = CA².

But if I assume symmetry, then x2 = -x1, so let me check that case again. If x2 = -x1, then y2 = x2² / 8 = x1² / 8 = y1. Therefore, points B and C are (x1, y1) and (-x1, y1). Then, as before, AB² = x1² + y1², BC² = ( -x1 - x1 )² + 0 = ( -2x1 )² = 4x1². Therefore, AB² = BC² implies x1² + y1² = 4x1² => y1² = 3x1². But since y1 = x1² / 8, substituting:

(x1² / 8)^2 = 3x1² => x1^4 / 64 = 3x1² => x1^4 - 192x1² = 0 => x1²(x1² - 192) = 0. So x1² = 192, so x1 = ±√192 = ±8√3. Therefore, same result as before. Therefore, the side length is sqrt(AB²) = sqrt(4x1²) = 2x1. Wait, but earlier I had AB = 16√3. Wait, if x1 = 8√3, then 2x1 = 16√3. But AB² was calculated as 768, which is (16√3)^2 = 256 * 3 = 768. So that's correct.

But here's a thought: if I don't assume symmetry, is there another solution? Let me try to explore that.

Suppose that points B and C are not symmetric. Let me denote B as (x1, x1² / 8) and C as (x2, x2² / 8). Then, the distances AB, AC, BC must be equal.

So AB² = x1² + (x1² / 8)^2.

Similarly, AC² = x2² + (x2² / 8)^2.

And BC² = (x2 - x1)^2 + (x2² / 8 - x1² / 8)^2.

Since AB = AC, then AB² = AC²:

x1² + (x1^4)/64 = x2² + (x2^4)/64.

Which implies x1² - x2² + (x1^4 - x2^4)/64 = 0.

Factor x1^4 - x2^4 as (x1² - x2²)(x1² + x2²). Therefore:

(x1² - x2²)[1 + (x1² + x2²)/64] = 0.

Therefore, either x1² - x2² = 0, which implies x1 = ±x2, or the other factor is zero:

1 + (x1² + x2²)/64 = 0. But since x1² and x2² are non-negative, this sum is always ≥ 1, so this can't be zero. Therefore, the only possibility is x1² = x2², so x2 = ±x1.

Therefore, if AB = AC, then x2 = ±x1. If x2 = x1, then points B and C coincide, which is not allowed. Therefore, x2 = -x1. Therefore, the only possibility is that points B and C are symmetric with respect to the y-axis. Therefore, my initial assumption was correct. Therefore, the triangle must be symmetric with respect to the y-axis. Therefore, the side length is 16√3. Therefore, is that the answer?

Wait, but let me check again. Wait, if AB = BC, then we have AB² = BC², which led us to the equation that gave us x1² = 192. Therefore, the side length is 16√3. However, I have a doubt because I remember that in some cases, there might be another equilateral triangle with a different side length. Let me check with coordinates.

Alternatively, maybe I can parametrize the points differently. Let me think. Let me consider parametric equations for the parabola. For the parabola x² = 4py, parametric equations are x = 2pt, y = pt². In our case, 4p = 8, so p = 2. Therefore, parametric equations are x = 4t, y = 2t². Wait, let me check:

If x = 2pt, then with p = 2, x = 4t. And y = pt² = 2t². Therefore, yes, parametric equations are x = 4t, y = 2t². Therefore, any point on the parabola can be written as (4t, 2t²). Therefore, maybe using parameters t1 and t2 for points B and C.

Let me try this approach. Let me denote point A as (0,0), point B as (4t, 2t²), and point C as (4s, 2s²). Since the triangle is equilateral, the distances AB, BC, and CA must be equal.

First, compute AB²: (4t - 0)^2 + (2t² - 0)^2 = 16t² + 4t^4.

Similarly, AC²: 16s² + 4s^4.

BC²: (4s - 4t)^2 + (2s² - 2t²)^2 = 16(s - t)^2 + 4(s² - t²)^2.

Since AB = AC, then AB² = AC²:

16t² + 4t^4 = 16s² + 4s^4.

Which simplifies to 4t^4 - 4s^4 + 16t² - 16s² = 0 => 4(t^4 - s^4) + 16(t² - s²) = 0 => 4(t² - s²)(t² + s²) + 16(t² - s²) = 0 => (t² - s²)[4(t² + s²) + 16] = 0.

Therefore, either t² - s² = 0 => t = ±s, or 4(t² + s²) + 16 = 0. The second equation is impossible because t² + s² ≥ 0, so 4(...) + 16 ≥ 16 > 0. Therefore, only possibility is t² = s² => s = ±t. Again, if s = t, points coincide, so s = -t. Therefore, points B and C are (4t, 2t²) and (-4t, 2t²). Therefore, same as before. Therefore, this confirms the previous result.

Therefore, proceeding with s = -t, then compute AB² = 16t² + 4t^4, and BC² = [4(-t) - 4t]^2 + [2(-t)^2 - 2t²]^2 = (-8t)^2 + [2t² - 2t²]^2 = 64t² + 0 = 64t². Therefore, AB² = BC² implies 16t² + 4t^4 = 64t² => 4t^4 - 48t² = 0 => 4t²(t² - 12) = 0. Therefore, t² = 0 or t² = 12. Again, t² = 0 gives degenerate triangle, so t² = 12 => t = ±2√3. Therefore, points B and C are (4t, 2t²) = (4*(±2√3), 2*(12)) = (±8√3, 24). Therefore, same as before. Therefore, AB² = 16t² + 4t^4 = 16*12 + 4*144 = 192 + 576 = 768. Therefore, AB = sqrt(768) = 16√3. Therefore, same result. Therefore, the side length is 16√3. Therefore, is this the answer?

Wait, but I have a doubt because I remember that in some cases, there might be another equilateral triangle with a different orientation. For example, maybe not symmetric with respect to the y-axis? But according to the previous analysis, if AB = AC, then points must be symmetric. However, maybe if we don't assume AB = AC, but instead consider AB = BC and BC = CA? Wait, but in the previous analysis, we saw that AB = AC is necessary for the triangle to be equilateral. Wait, no. Wait, in general, for a triangle to be equilateral, all sides must be equal. Therefore, AB = BC = CA. Therefore, if we don't assume AB = AC, but instead set AB = BC and BC = CA, maybe there's another solution? Wait, but according to the previous parametrization, if AB = AC, then points must be symmetric. However, if we don't assume AB = AC, but instead set AB = BC and BC = CA, maybe there's another solution?

Wait, but in reality, if AB = BC and BC = CA, then AB = AC automatically. Therefore, the triangle is equilateral. Therefore, the previous analysis holds. Therefore, the only solution is the symmetric one. Therefore, the side length is 16√3. Therefore, maybe that is the answer. However, let me check with coordinates.

Alternatively, maybe I can use complex numbers or rotation to find the coordinates. Let me think. Since the triangle is equilateral, rotating point B around point A by 60 degrees should give point C or something like that. Let me try this approach.

Let me consider point A at (0,0), point B at (x, y), then point C can be obtained by rotating point B around A by 60 degrees. The rotation matrix for 60 degrees is [cosθ -sinθ; sinθ cosθ], where θ = 60°, so cosθ = 0.5, sinθ = (√3)/2. Therefore, if point B is (x, y), then point C would be (x*cosθ - y*sinθ, x*sinθ + y*cosθ). Therefore, coordinates of C would be (0.5x - (√3/2)y, (√3/2)x + 0.5y). Since point C lies on the parabola x² = 8y, so its coordinates must satisfy that equation. Also, point B lies on the parabola, so x² = 8y. Therefore, we can write equations for both points.

Let me denote point B as (x, x²/8). Then, point C is (0.5x - (√3/2)(x²/8), (√3/2)x + 0.5*(x²/8)). Let me compute coordinates of C:

First coordinate: 0.5x - (√3/2)(x²/8) = (x/2) - (√3 x²)/16.

Second coordinate: (√3/2)x + (x²)/16.

Since point C lies on the parabola, its coordinates must satisfy:

[ (x/2 - (√3 x²)/16 ) ]² = 8 * [ (√3/2)x + (x²)/16 ].

This seems complicated, but let me try to compute it.

Let me denote:

Let me compute left side: [ (x/2 - (√3 x²)/16 ) ]².

Let me factor x/2:

= [ x/2 (1 - (√3 x)/8 ) ]² = (x² / 4)(1 - (√3 x)/8 )².

Right side: 8 * [ (√3/2)x + (x²)/16 ] = 8*(√3/2 x) + 8*(x² /16) = 4√3 x + 0.5 x².

Therefore, equation becomes:

(x² / 4)(1 - (√3 x)/8 )² = 4√3 x + 0.5 x².

This seems messy, but let me expand the left side.

First, expand (1 - (√3 x)/8 )²:

= 1 - 2*(√3 x)/8 + ( (√3 x)/8 )² = 1 - (√3 x)/4 + (3x²)/64.

Therefore, left side:

(x² / 4)[1 - (√3 x)/4 + (3x²)/64] = (x² / 4) - (x² * √3 x)/16 + (3x^4)/256.

Simplify:

= (x²)/4 - (√3 x³)/16 + (3x^4)/256.

Therefore, equation:

(x²)/4 - (√3 x³)/16 + (3x^4)/256 = 4√3 x + 0.5 x².

Bring all terms to left side:

(x²)/4 - (√3 x³)/16 + (3x^4)/256 - 4√3 x - 0.5 x² = 0.

Combine like terms:

(x²)/4 - 0.5x² = (x²)/4 - 2x²/4 = (-x²)/4.

Therefore:

- x² / 4 - (√3 x³)/16 + (3x^4)/256 - 4√3 x = 0.

Multiply both sides by 256 to eliminate denominators:

-64x² - 16√3 x³ + 3x^4 - 1024√3 x = 0.

Rearranged:

3x^4 - 16√3 x³ - 64x² - 1024√3 x = 0.

Factor out x:

x(3x³ - 16√3 x² - 64x - 1024√3 ) = 0.

So solutions are x = 0, which gives degenerate triangle, or solving 3x³ - 16√3 x² - 64x - 1024√3 = 0.

This seems very complicated. Maybe there is a rational root? Let me try to see if x = 8√3 is a root. Let me substitute x = 8√3 into the cubic:

First compute each term:

3x³ = 3*(8√3)^3 = 3*(512 * 3√3) = 3*1536√3 = 4608√3.

-16√3 x² = -16√3*(8√3)^2 = -16√3*(64*3) = -16√3*192 = -3072√3.

-64x = -64*(8√3) = -512√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.

Wow! So x = 8√3 is a root. Therefore, we can factor (x - 8√3) from the cubic.

Let me perform polynomial division or use synthetic division. Let me denote the cubic as:

3x³ - 16√3 x² - 64x - 1024√3.

Let me write it as 3x³ - 16√3 x² - 64x - 1024√3.

Let me factor out (x - 8√3). Let me use polynomial division.

Alternatively, since we know that x = 8√3 is a root, let me write the cubic as (x - 8√3)(Ax² + Bx + C). Let me find A, B, C.

Multiply out:

(x - 8√3)(Ax² + Bx + C) = Ax³ + Bx² + Cx - 8√3 Ax² - 8√3 Bx - 8√3 C.

Combine like terms:

Ax³ + (B - 8√3 A)x² + (C - 8√3 B)x - 8√3 C.

Set equal to original cubic:

3x³ - 16√3 x² - 64x - 1024√3.

Therefore, equate coefficients:

A = 3.

B - 8√3 A = -16√3 => B - 8√3 * 3 = -16√3 => B - 24√3 = -16√3 => B = 8√3.

C - 8√3 B = -64 => C - 8√3 * 8√3 = -64 => C - 8*8*(√3)^2 = -64 => C - 64*3 = -64 => C - 192 = -64 => C = 128.

Then, the last term: -8√3 C = -1024√3 => -8√3 * 128 = -1024√3. Let me check:

-8*128 = -1024, so yes, -1024√3. Correct.

Therefore, the cubic factors as (x - 8√3)(3x² + 8√3 x + 128).

Now, set 3x² + 8√3 x + 128 = 0. Let me check discriminant:

Discriminant D = (8√3)^2 - 4*3*128 = 192 - 1536 = -1344 < 0. Therefore, no real roots. Therefore, the only real solution is x = 8√3. Therefore, the only non-degenerate solution is x = 8√3. Therefore, this confirms our previous result. Therefore, the coordinates of point B are (8√3, 24), and point C is (-8√3, 24). Therefore, the side length is 16√3. Therefore, the answer is 16√3. Therefore, I think this is correct.

But let me check once more with another approach. Let me compute the angles. For example, in the triangle with vertices at (0,0), (8√3, 24), (-8√3, 24). Let me compute the vectors AB and AC. Vector AB is (8√3, 24), vector AC is (-8√3, 24). The angle between AB and AC can be found using the dot product.

Dot product AB · AC = (8√3)(-8√3) + (24)(24) = -64*3 + 576 = -192 + 576 = 384.

The magnitude of AB and AC is 16√3, as before. Therefore, the cosine of the angle between AB and AC is (384)/( (16√3)^2 ) = 384 / (256 * 3) = 384 / 768 = 0.5. Therefore, the angle is arccos(0.5) = 60 degrees. Therefore, the angle at A is 60 degrees, which is consistent with an equilateral triangle. Therefore, this confirms that the triangle is indeed equilateral. Therefore, the side length is 16√3.

But let me check if there is another triangle where the angle is 60 degrees but not symmetric. However, according to previous analysis, the only real solution is the symmetric one. Therefore, I think this is the only possible triangle. Therefore, the answer is 16√3.

But wait, let me check with another parametrization. Suppose I take a general point on the parabola, say (x, x²/8), and then find another point such that the triangle is equilateral. But since we already did that with rotation and found only one solution, I think that's the only one.

Alternatively, maybe there's another triangle where the vertex at the origin is not between the other two points? Wait, but in a parabola, the vertex is the lowest point. If we have an equilateral triangle with one vertex at the origin, the other two points must be above the origin. So the triangle is above the vertex. So the configuration we found is the only one.

Alternatively, maybe there's a triangle where one vertex is at the origin, and the other two are on different sides? But since the parabola is symmetric, I think the symmetric case is the only one.

Therefore, after multiple approaches and checks, I think the side length is indeed 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 $, with one vertex at the vertex of the parabola (which is at the origin, $ (0, 0) $), we proceed as follows:

---

### **Step 1: Understand the Geometry**

The parabola $ x^2 = 8y $ is symmetric about the y-axis. The vertex of the parabola is at the origin $ (0, 0) $. We are to inscribe an equilateral triangle in this parabola such that one of its vertices is at the origin.

Let the other two vertices of the triangle be $ B = (x, y) $ and $ C = (-x, y) $, due to the symmetry of the parabola. Since both points lie on the parabola, they must satisfy the equation:

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

So, the coordinates of the points are:
- $ A = (0, 0) $
- $ B = (x, \frac{x^2}{8}) $
- $ C = (-x, \frac{x^2}{8}) $

---

### **Step 2: Use the Equilateral Triangle Condition**

In an equilateral triangle, all sides are equal. So we compute the distances:

- $ AB = \sqrt{x^2 + \left(\frac{x^2}{8}\right)^2} = \sqrt{x^2 + \frac{x^4}{64}} $
- $ BC = \sqrt{(x - (-x))^2 + \left(\frac{x^2}{8} - \frac{x^2}{8}\right)^2} = \sqrt{(2x)^2} = 2|x| $

Set $ AB = BC $:

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

Squaring both sides:

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

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

Multiply both sides by 64:

$$
x^4 = 192x^2
$$

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

So, $ x^2 = 0 $ or $ x^2 = 192 $. Since $ x = 0 $ would collapse the triangle, we take $ x^2 = 192 $, so $ x = \pm 8\sqrt{3} $.

---

### **Step 3: Compute the Side Length**

Using $ x = 8\sqrt{3} $, the coordinates of the points are:

- $ A = (0, 0) $
- $ B = (8\sqrt{3}, 24) $
- $ C = (-8\sqrt{3}, 24) $

Now compute the side length $ AB $:

$$
AB = \sqrt{(8\sqrt{3})^2 + 24^2} = \sqrt{192 + 576} = \sqrt{768} = 16\sqrt{3}
$$

Since the triangle is equilateral, all sides are equal, so the side length is:

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

---

### **Final Answer**

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

[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, 4p = 8, so p = 2. That means the focus is at (0, 2), but maybe that's not directly relevant here.

Now, the problem says an equilateral triangle is inscribed in the parabola, with one vertex at the vertex of the parabola, which is (0,0). Let me denote the three vertices of the triangle as A, B, and C. Let's say point A is at (0,0). Then points B and C are somewhere on the parabola.

Since it's an equilateral triangle, all sides must be equal. So AB = BC = CA. But since A is at (0,0), I need to find coordinates for B and C on the parabola such that the distances AB, BC, and CA are equal.

Let me denote coordinates for points B and C. Let me assume that points B and C are symmetric with respect to the y-axis. Because the parabola is symmetric about the y-axis, and if we have an equilateral triangle with one vertex at the origin, maybe the other two vertices are symmetric. That might simplify things.

So let me suppose that point B is (a, b) and point C is (-a, b), since they are symmetric across the y-axis. Then, since they lie on the parabola $ x^2 = 8y $, their coordinates must satisfy this equation. Therefore, for point B: $ a^2 = 8b $, so $ b = a^2 / 8 $. Similarly for point C, same thing.

Now, let me compute the distances between these points. First, AB and AC. Since A is (0,0), the distance AB is sqrt[(a - 0)^2 + (b - 0)^2] = sqrt(a² + b²). Similarly, AC is the same because of symmetry.

Then, the distance BC is between (a, b) and (-a, b). That distance is sqrt[(-a - a)^2 + (b - b)^2] = sqrt[(-2a)^2 + 0] = sqrt[4a²] = 2|a|. Since a is a coordinate on the parabola, it can be positive or negative, but since we squared it, we can assume a is positive without loss of generality.

So, since the triangle is equilateral, AB = BC. Therefore:

sqrt(a² + b²) = 2a.

But we also know that b = a² / 8. Let me substitute that into the equation:

sqrt(a² + (a² / 8)^2) = 2a.

Let me square both sides to eliminate the square root:

a² + (a^4) / 64 = 4a².

Simplify:

(a^4)/64 + a² - 4a² = 0

Which becomes:

(a^4)/64 - 3a² = 0

Multiply both sides by 64 to eliminate the denominator:

a^4 - 192a² = 0

Factor:

a²(a² - 192) = 0

So, solutions are a² = 0 or a² = 192. Since a = 0 would give points B and C both at (0,0), which would collapse the triangle, we discard a² = 0. Therefore, a² = 192, so a = sqrt(192) or -sqrt(192). Since we assumed a positive, a = sqrt(192).

Simplify sqrt(192): sqrt(64*3) = 8*sqrt(3). Therefore, a = 8√3. Then, b = a² / 8 = (192)/8 = 24. Therefore, points B and C are (8√3, 24) and (-8√3, 24).

Now, let me check if the distance AB equals BC. Let me compute AB:

sqrt[(8√3)^2 + (24)^2] = sqrt(64*3 + 576) = sqrt(192 + 576) = sqrt(768). Simplify sqrt(768): sqrt(256*3) = 16√3.

Distance BC is 2a = 2*(8√3) = 16√3. So yes, AB = BC. But wait, the triangle is supposed to be equilateral, so we also need to check that AC = AB and BC = AC, which they are by symmetry. But also, we need to check that the third side, which is AC, is equal. Wait, AC is same as AB, so that's already covered. However, we also need to check that the distance between B and C is equal to AB. Which we did, and it is. So that seems okay.

But wait, hold on. Wait, in an equilateral triangle, all sides are equal. So AB = BC = CA. But here, we have AB = BC, and CA is same as AB. So that's okay. But wait, but maybe there's a mistake here? Because if points B and C are symmetric across the y-axis, then the triangle is isoceles with AB = AC, but for it to be equilateral, BC must also equal AB. Which we have done. So according to this, the side length is 16√3. But let me check if that's correct.

Wait, but let me think again. If the triangle is equilateral, then all angles are 60 degrees. However, in this case, with points at (0,0), (8√3, 24), (-8√3, 24), is the angle at A 60 degrees? Let me check.

Alternatively, maybe there's a mistake in assuming that points B and C are symmetric across the y-axis. Maybe that's not the case? Wait, but given the symmetry of the parabola, maybe that's the only way? Or maybe there's another configuration?

Wait, but let me think again. Suppose that the triangle is not symmetric. Maybe the other two vertices are not symmetric? But then how would we find them? It might be more complicated. However, given that the parabola is symmetric about the y-axis, and the triangle has one vertex at the origin, it's plausible that the other two vertices are symmetric. But maybe there's another triangle where they are not symmetric? But since the problem says "an equilateral triangle", maybe there are multiple such triangles, but the question is to find the side length. However, maybe the symmetric case is the only one? Or maybe not?

Wait, but let me check if the triangle I found is actually equilateral. Let me compute all sides.

AB: distance from (0,0) to (8√3, 24): sqrt[(8√3)^2 + 24^2] = sqrt(192 + 576) = sqrt(768) = 16√3.

BC: distance between (8√3, 24) and (-8√3, 24): sqrt[( -8√3 - 8√3)^2 + (24 - 24)^2] = sqrt[(-16√3)^2 + 0] = sqrt(768) = 16√3.

AC: same as AB, so 16√3. So yes, all sides are equal. Therefore, this is an equilateral triangle. Therefore, the side length is 16√3. But wait, let me check if this is correct. However, I have a feeling that maybe there's a mistake here because I remember that in some cases, when dealing with parabolas and equilateral triangles, there might be another configuration. Let me think again.

Alternatively, maybe I need to consider that the triangle is not symmetric? Let me try to approach this more generally.

Let me suppose that the three vertices are A(0,0), B(x1, y1), and C(x2, y2), all lying on the parabola x² = 8y. Therefore, y1 = x1² / 8 and y2 = x2² / 8.

Since the triangle is equilateral, the distances AB, BC, and CA must be equal. So:

AB² = (x1 - 0)^2 + (y1 - 0)^2 = x1² + y1² = x1² + (x1² / 8)^2 = x1² + x1^4 / 64.

Similarly, AC² = x2² + (x2² / 8)^2.

And BC² = (x2 - x1)^2 + (y2 - y1)^2 = (x2 - x1)^2 + (x2² / 8 - x1² / 8)^2.

Since AB = BC = CA, so AB² = BC² and AB² = CA².

But if I assume symmetry, then x2 = -x1, so let me check that case again. If x2 = -x1, then y2 = x2² / 8 = x1² / 8 = y1. Therefore, points B and C are (x1, y1) and (-x1, y1). Then, as before, AB² = x1² + y1², BC² = ( -x1 - x1 )² + 0 = ( -2x1 )² = 4x1². Therefore, AB² = BC² implies x1² + y1² = 4x1² => y1² = 3x1². But since y1 = x1² / 8, substituting:

(x1² / 8)^2 = 3x1² => x1^4 / 64 = 3x1² => x1^4 - 192x1² = 0 => x1²(x1² - 192) = 0. So x1² = 192, so x1 = ±√192 = ±8√3. Therefore, same result as before. Therefore, the side length is sqrt(AB²) = sqrt(4x1²) = 2x1. Wait, but earlier I had AB = 16√3. Wait, if x1 = 8√3, then 2x1 = 16√3. But AB² was calculated as 768, which is (16√3)^2 = 256 * 3 = 768. So that's correct.

But here's a thought: if I don't assume symmetry, is there another solution? Let me try to explore that.

Suppose that points B and C are not symmetric. Let me denote B as (x1, x1² / 8) and C as (x2, x2² / 8). Then, the distances AB, AC, BC must be equal.

So AB² = x1² + (x1² / 8)^2.

Similarly, AC² = x2² + (x2² / 8)^2.

And BC² = (x2 - x1)^2 + (x2² / 8 - x1² / 8)^2.

Since AB = AC, then AB² = AC²:

x1² + (x1^4)/64 = x2² + (x2^4)/64.

Which implies x1² - x2² + (x1^4 - x2^4)/64 = 0.

Factor x1^4 - x2^4 as (x1² - x2²)(x1² + x2²). Therefore:

(x1² - x2²)[1 + (x1² + x2²)/64] = 0.

Therefore, either x1² - x2² = 0, which implies x1 = ±x2, or the other factor is zero:

1 + (x1² + x2²)/64 = 0. But since x1² and x2² are non-negative, this sum is always ≥ 1, so this can't be zero. Therefore, the only possibility is x1² = x2², so x2 = ±x1.

Therefore, if AB = AC, then x2 = ±x1. If x2 = x1, then points B and C coincide, which is not allowed. Therefore, x2 = -x1. Therefore, the only possibility is that points B and C are symmetric with respect to the y-axis. Therefore, my initial assumption was correct. Therefore, the triangle must be symmetric with respect to the y-axis. Therefore, the side length is 16√3. Therefore, is that the answer?

Wait, but let me check again. Wait, if AB = BC, then we have AB² = BC², which led us to the equation that gave us x1² = 192. Therefore, the side length is 16√3. However, I have a doubt because I remember that in some cases, there might be another equilateral triangle with a different side length. Let me check with coordinates.

Alternatively, maybe I can parametrize the points differently. Let me think. Let me consider parametric equations for the parabola. For the parabola x² = 4py, parametric equations are x = 2pt, y = pt². In our case, 4p = 8, so p = 2. Therefore, parametric equations are x = 4t, y = 2t². Wait, let me check:

If x = 2pt, then with p = 2, x = 4t. And y = pt² = 2t². Therefore, yes, parametric equations are x = 4t, y = 2t². Therefore, any point on the parabola can be written as (4t, 2t²). Therefore, maybe using parameters t1 and t2 for points B and C.

Let me try this approach. Let me denote point A as (0,0), point B as (4t, 2t²), and point C as (4s, 2s²). Since the triangle is equilateral, the distances AB, BC, and CA must be equal.

First, compute AB²: (4t - 0)^2 + (2t² - 0)^2 = 16t² + 4t^4.

Similarly, AC²: 16s² + 4s^4.

BC²: (4s - 4t)^2 + (2s² - 2t²)^2 = 16(s - t)^2 + 4(s² - t²)^2.

Since AB = AC, then AB² = AC²:

16t² + 4t^4 = 16s² + 4s^4.

Which simplifies to 4t^4 - 4s^4 + 16t² - 16s² = 0 => 4(t^4 - s^4) + 16(t² - s²) = 0 => 4(t² - s²)(t² + s²) + 16(t² - s²) = 0 => (t² - s²)[4(t² + s²) + 16] = 0.

Therefore, either t² - s² = 0 => t = ±s, or 4(t² + s²) + 16 = 0. The second equation is impossible because t² + s² ≥ 0, so 4(...) + 16 ≥ 16 > 0. Therefore, only possibility is t² = s² => s = ±t. Again, if s = t, points coincide, so s = -t. Therefore, points B and C are (4t, 2t²) and (-4t, 2t²). Therefore, same as before. Therefore, this confirms the previous result.

Therefore, proceeding with s = -t, then compute AB² = 16t² + 4t^4, and BC² = [4(-t) - 4t]^2 + [2(-t)^2 - 2t²]^2 = (-8t)^2 + [2t² - 2t²]^2 = 64t² + 0 = 64t². Therefore, AB² = BC² implies 16t² + 4t^4 = 64t² => 4t^4 - 48t² = 0 => 4t²(t² - 12) = 0. Therefore, t² = 0 or t² = 12. Again, t² = 0 gives degenerate triangle, so t² = 12 => t = ±2√3. Therefore, points B and C are (4t, 2t²) = (4*(±2√3), 2*(12)) = (±8√3, 24). Therefore, same as before. Therefore, AB² = 16t² + 4t^4 = 16*12 + 4*144 = 192 + 576 = 768. Therefore, AB = sqrt(768) = 16√3. Therefore, same result. Therefore, the side length is 16√3. Therefore, is this the answer?

Wait, but I have a doubt because I remember that in some cases, there might be another equilateral triangle with a different orientation. For example, maybe not symmetric with respect to the y-axis? But according to the previous analysis, if AB = AC, then points must be symmetric. However, maybe if we don't assume AB = AC, but instead consider AB = BC and BC = CA? Wait, but in the previous analysis, we saw that AB = AC is necessary for the triangle to be equilateral. Wait, no. Wait, in general, for a triangle to be equilateral, all sides must be equal. Therefore, AB = BC = CA. Therefore, if we don't assume AB = AC, but instead set AB = BC and BC = CA, maybe there's another solution? Wait, but according to the previous parametrization, if AB = AC, then points must be symmetric. However, if we don't assume AB = AC, but instead set AB = BC and BC = CA, maybe there's another solution?

Wait, but in reality, if AB = BC and BC = CA, then AB = AC automatically. Therefore, the triangle is equilateral. Therefore, the previous analysis holds. Therefore, the only solution is the symmetric one. Therefore, the side length is 16√3. Therefore, maybe that is the answer. However, let me check with coordinates.

Alternatively, maybe I can use complex numbers or rotation to find the coordinates. Let me think. Since the triangle is equilateral, rotating point B around point A by 60 degrees should give point C or something like that. Let me try this approach.

Let me consider point A at (0,0), point B at (x, y), then point C can be obtained by rotating point B around A by 60 degrees. The rotation matrix for 60 degrees is [cosθ -sinθ; sinθ cosθ], where θ = 60°, so cosθ = 0.5, sinθ = (√3)/2. Therefore, if point B is (x, y), then point C would be (x*cosθ - y*sinθ, x*sinθ + y*cosθ). Therefore, coordinates of C would be (0.5x - (√3/2)y, (√3/2)x + 0.5y). Since point C lies on the parabola x² = 8y, so its coordinates must satisfy that equation. Also, point B lies on the parabola, so x² = 8y. Therefore, we can write equations for both points.

Let me denote point B as (x, x²/8). Then, point C is (0.5x - (√3/2)(x²/8), (√3/2)x + 0.5*(x²/8)). Let me compute coordinates of C:

First coordinate: 0.5x - (√3/2)(x²/8) = (x/2) - (√3 x²)/16.

Second coordinate: (√3/2)x + (x²)/16.

Since point C lies on the parabola, its coordinates must satisfy:

[ (x/2 - (√3 x²)/16 ) ]² = 8 * [ (√3/2)x + (x²)/16 ].

This seems complicated, but let me try to compute it.

Let me denote:

Let me compute left side: [ (x/2 - (√3 x²)/16 ) ]².

Let me factor x/2:

= [ x/2 (1 - (√3 x)/8 ) ]² = (x² / 4)(1 - (√3 x)/8 )².

Right side: 8 * [ (√3/2)x + (x²)/16 ] = 8*(√3/2 x) + 8*(x² /16) = 4√3 x + 0.5 x².

Therefore, equation becomes:

(x² / 4)(1 - (√3 x)/8 )² = 4√3 x + 0.5 x².

This seems messy, but let me expand the left side.

First, expand (1 - (√3 x)/8 )²:

= 1 - 2*(√3 x)/8 + ( (√3 x)/8 )² = 1 - (√3 x)/4 + (3x²)/64.

Therefore, left side:

(x² / 4)[1 - (√3 x)/4 + (3x²)/64] = (x² / 4) - (x² * √3 x)/16 + (3x^4)/256.

Simplify:

= (x²)/4 - (√3 x³)/16 + (3x^4)/256.

Therefore, equation:

(x²)/4 - (√3 x³)/16 + (3x^4)/256 = 4√3 x + 0.5 x².

Bring all terms to left side:

(x²)/4 - (√3 x³)/16 + (3x^4)/256 - 4√3 x - 0.5 x² = 0.

Combine like terms:

(x²)/4 - 0.5x² = (x²)/4 - 2x²/4 = (-x²)/4.

Therefore:

- x² / 4 - (√3 x³)/16 + (3x^4)/256 - 4√3 x = 0.

Multiply both sides by 256 to eliminate denominators:

-64x² - 16√3 x³ + 3x^4 - 1024√3 x = 0.

Rearranged:

3x^4 - 16√3 x³ - 64x² - 1024√3 x = 0.

Factor out x:

x(3x³ - 16√3 x² - 64x - 1024√3 ) = 0.

So solutions are x = 0, which gives degenerate triangle, or solving 3x³ - 16√3 x² - 64x - 1024√3 = 0.

This seems very complicated. Maybe there is a rational root? Let me try to see if x = 8√3 is a root. Let me substitute x = 8√3 into the cubic:

First compute each term:

3x³ = 3*(8√3)^3 = 3*(512 * 3√3) = 3*1536√3 = 4608√3.

-16√3 x² = -16√3*(8√3)^2 = -16√3*(64*3) = -16√3*192 = -3072√3.

-64x = -64*(8√3) = -512√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.

Wow! So x = 8√3 is a root. Therefore, we can factor (x - 8√3) from the cubic.

Let me perform polynomial division or use synthetic division. Let me denote the cubic as:

3x³ - 16√3 x² - 64x - 1024√3.

Let me write it as 3x³ - 16√3 x² - 64x - 1024√3.

Let me factor out (x - 8√3). Let me use polynomial division.

Alternatively, since we know that x = 8√3 is a root, let me write the cubic as (x - 8√3)(Ax² + Bx + C). Let me find A, B, C.

Multiply out:

(x - 8√3)(Ax² + Bx + C) = Ax³ + Bx² + Cx - 8√3 Ax² - 8√3 Bx - 8√3 C.

Combine like terms:

Ax³ + (B - 8√3 A)x² + (C - 8√3 B)x - 8√3 C.

Set equal to original cubic:

3x³ - 16√3 x² - 64x - 1024√3.

Therefore, equate coefficients:

A = 3.

B - 8√3 A = -16√3 => B - 8√3 * 3 = -16√3 => B - 24√3 = -16√3 => B = 8√3.

C - 8√3 B = -64 => C - 8√3 * 8√3 = -64 => C - 8*8*(√3)^2 = -64 => C - 64*3 = -64 => C - 192 = -64 => C = 128.

Then, the last term: -8√3 C = -1024√3 => -8√3 * 128 = -1024√3. Let me check:

-8*128 = -1024, so yes, -1024√3. Correct.

Therefore, the cubic factors as (x - 8√3)(3x² + 8√3 x + 128).

Now, set 3x² + 8√3 x + 128 = 0. Let me check discriminant:

Discriminant D = (8√3)^2 - 4*3*128 = 192 - 1536 = -1344 < 0. Therefore, no real roots. Therefore, the only real solution is x = 8√3. Therefore, the only non-degenerate solution is x = 8√3. Therefore, this confirms our previous result. Therefore, the coordinates of point B are (8√3, 24), and point C is (-8√3, 24). Therefore, the side length is 16√3. Therefore, the answer is 16√3. Therefore, I think this is correct.

But let me check once more with another approach. Let me compute the angles. For example, in the triangle with vertices at (0,0), (8√3, 24), (-8√3, 24). Let me compute the vectors AB and AC. Vector AB is (8√3, 24), vector AC is (-8√3, 24). The angle between AB and AC can be found using the dot product.

Dot product AB · AC = (8√3)(-8√3) + (24)(24) = -64*3 + 576 = -192 + 576 = 384.

The magnitude of AB and AC is 16√3, as before. Therefore, the cosine of the angle between AB and AC is (384)/( (16√3)^2 ) = 384 / (256 * 3) = 384 / 768 = 0.5. Therefore, the angle is arccos(0.5) = 60 degrees. Therefore, the angle at A is 60 degrees, which is consistent with an equilateral triangle. Therefore, this confirms that the triangle is indeed equilateral. Therefore, the side length is 16√3.

But let me check if there is another triangle where the angle is 60 degrees but not symmetric. However, according to previous analysis, the only real solution is the symmetric one. Therefore, I think this is the only possible triangle. Therefore, the answer is 16√3.

But wait, let me check with another parametrization. Suppose I take a general point on the parabola, say (x, x²/8), and then find another point such that the triangle is equilateral. But since we already did that with rotation and found only one solution, I think that's the only one.

Alternatively, maybe there's another triangle where the vertex at the origin is not between the other two points? Wait, but in a parabola, the vertex is the lowest point. If we have an equilateral triangle with one vertex at the origin, the other two points must be above the origin. So the triangle is above the vertex. So the configuration we found is the only one.

Alternatively, maybe there's a triangle where one vertex is at the origin, and the other two are on different sides? But since the parabola is symmetric, I think the symmetric case is the only one.

Therefore, after multiple approaches and checks, I think the side length is indeed 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 $, with one vertex at the vertex of the parabola (which is at the origin, $ (0, 0) $), we proceed as follows:

---

### **Step 1: Understand the Geometry**

The parabola $ x^2 = 8y $ is symmetric about the y-axis. The vertex of the parabola is at the origin $ (0, 0) $. We are to inscribe an equilateral triangle in this parabola such that one of its vertices is at the origin.

Let the other two vertices of the triangle be $ B = (x, y) $ and $ C = (-x, y) $, due to the symmetry of the parabola. Since both points lie on the parabola, they must satisfy the equation:

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

So, the coordinates of the points are:
- $ A = (0, 0) $
- $ B = (x, \frac{x^2}{8}) $
- $ C = (-x, \frac{x^2}{8}) $

---

### **Step 2: Use the Equilateral Triangle Condition**

In an equilateral triangle, all sides are equal. So we compute the distances:

- $ AB = \sqrt{x^2 + \left(\frac{x^2}{8}\right)^2} = \sqrt{x^2 + \frac{x^4}{64}} $
- $ BC = \sqrt{(x - (-x))^2 + \left(\frac{x^2}{8} - \frac{x^2}{8}\right)^2} = \sqrt{(2x)^2} = 2|x| $

Set $ AB = BC $:

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

Squaring both sides:

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

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

Multiply both sides by 64:

$$
x^4 = 192x^2
$$

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

So, $ x^2 = 0 $ or $ x^2 = 192 $. Since $ x = 0 $ would collapse the triangle, we take $ x^2 = 192 $, so $ x = \pm 8\sqrt{3} $.

---

### **Step 3: Compute the Side Length**

Using $ x = 8\sqrt{3} $, the coordinates of the points are:

- $ A = (0, 0) $
- $ B = (8\sqrt{3}, 24) $
- $ C = (-8\sqrt{3}, 24) $

Now compute the side length $ AB $:

$$
AB = \sqrt{(8\sqrt{3})^2 + 24^2} = \sqrt{192 + 576} = \sqrt{768} = 16\sqrt{3}
$$

Since the triangle is equilateral, all sides are equal, so the side length is:

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

---

### **Final Answer**

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

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 after reflecting each point on the circle x² + y² = 25 over the point (4,1). The final equation is supposed to 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.

Reflecting a point P over a point Q results in a point P' such that Q is the midpoint of the segment PP'. So, if I have a point (x, y) on the original circle, its reflection (let's call it (x', y')) over (4,1) should satisfy the midpoint formula. That is, the midpoint between (x, y) and (x', y') is (4,1).

So, the midpoint formula says that:

( (x + x') / 2 , (y + y') / 2 ) = (4, 1)

Therefore, solving for x' and y', we get:

x + x' = 8 => x' = 8 - x

y + y' = 2 => y' = 2 - y

So, the reflection of any point (x, y) over (4,1) is (8 - x, 2 - y).

Now, since the original points lie on the circle x² + y² = 25, their reflections will satisfy some equation. Let me denote the reflected point as (X, Y). So, X = 8 - x and Y = 2 - y. Therefore, we can express x and y in terms of X and Y:

x = 8 - X

y = 2 - Y

Since (x, y) lies on the original circle, substituting these into the original equation:

(8 - X)² + (2 - Y)² = 25

Now, I need to expand this equation and rearrange it into the form x² + a y² + b x + c y + d = 0. Wait, but here the variables are X and Y, which are the coordinates of the image points. However, the problem says the set of image points satisfies the equation x² + a y² + b x + c y + d = 0. So, I need to write the equation in terms of X and Y, but since X and Y are just variables, we can just replace them with x and y for the equation. Let me check that.

So, expanding (8 - X)² + (2 - Y)² = 25:

First, expand (8 - X)²:

= 64 - 16X + X²

Then, expand (2 - Y)²:

= 4 - 4Y + Y²

Adding them together:

64 - 16X + X² + 4 - 4Y + Y² = 25

Combine like terms:

(64 + 4) + (-16X - 4Y) + (X² + Y²) = 25

Which is:

68 - 16X - 4Y + X² + Y² = 25

Now, subtract 25 from both sides to set the equation to zero:

X² + Y² - 16X - 4Y + 68 - 25 = 0

Calculating 68 - 25:

68 - 25 = 43

So, the equation becomes:

X² + Y² - 16X - 4Y + 43 = 0

But the problem states that the equation is in the form x² + a y² + b x + c y + d = 0. Comparing this with our equation, we can see that:

x² + y² - 16x - 4y + 43 = 0

Therefore, the coefficients are:

a = 1 (since the coefficient of y² is 1)

b = -16

c = -4

d = 43

Wait, but the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". So, in our case, the equation is x² + y² -16x -4y +43 = 0, so comparing term by term:

x² + (1)y² + (-16)x + (-4)y + 43 = 0. Therefore, the ordered quadruple is (a, b, c, d) = (1, -16, -4, 43).

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

First, reflection over a point: yes, midpoint formula. So if original point is (x, y), then reflection is (8 - x, 2 - y). Then substituting into original equation.

Original equation: x² + y² = 25. So replacing x with 8 - X and y with 2 - Y. Then expanding:

(8 - X)^2 + (2 - Y)^2 = 25.

Expanding:

(64 - 16X + X²) + (4 - 4Y + Y²) = 25.

Adding 64 + 4 = 68. Then -16X -4Y. Then X² + Y².

So 68 -16X -4Y + X² + Y² = 25.

Subtract 25: X² + Y² -16X -4Y + 43 = 0.

Yes, that seems correct.

But let me check if there's a possibility that the problem expects the equation in terms of the original variables? Wait, no. The image points are (X, Y), so the equation they satisfy is in terms of X and Y. But since the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0", so they just use x and y as variables for the image points. So replacing X and Y with x and y, the equation is x² + y² -16x -4y +43 = 0. Therefore, the coefficients are a=1, b=-16, c=-4, d=43.

But let me check if there's a different approach. Maybe I can think about the reflection of the circle. Reflecting a circle over a point... Well, reflecting a circle over a point results in another circle. The original circle has center at (0,0) and radius 5. Reflecting over the point (4,1) would result in a circle whose center is the reflection of the original center over (4,1).

Let me try this approach. The original center is (0,0). Reflecting (0,0) over (4,1) gives a point (8, 2), since midpoint between (0,0) and (8,2) is (4,1). Therefore, the reflected circle has center (8,2) and same radius 5. Therefore, the equation is (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 + 43 = 0.

Same result as before! So that confirms that the equation is indeed x² + y² -16x -4y +43 = 0. Therefore, the ordered quadruple is (1, -16, -4, 43).

But let me check once more if the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". So, in our case, the equation is x² + y² -16x -4y +43 = 0. Therefore, comparing term by term:

x² + (1)y² + (-16)x + (-4)y + 43 = 0. Therefore, a=1, b=-16, c=-4, d=43.

So the ordered quadruple is (1, -16, -4, 43).

But let me check if there's any possible mistake in the reflection. For example, if I had messed up the reflection formula. Let me think again. If you reflect a point (x, y) over the point (h, k), then the reflection point (x', y') is such that (h, k) is the midpoint. Therefore:

h = (x + x')/2 => x' = 2h - x

Similarly, y' = 2k - y.

So in this case, h = 4, k = 1. Therefore, x' = 8 - x, y' = 2 - y. Which is what I had before. So that's correct.

Therefore, substituting into original equation gives the equation of the reflected circle.

Alternatively, if I think of the reflected circle as having center (8, 2) and radius 5, then equation is (x - 8)^2 + (y - 2)^2 = 25. Expanding that gives x² -16x +64 + y² -4y +4 = 25 => x² + y² -16x -4y + 68 -25 = 0 => x² + y² -16x -4y +43 = 0.

Same result. So that's correct.

Therefore, the answer is (a, b, c, d) = (1, -16, -4, 43).

But let me check if the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". So, in our case, the equation is x² + y² -16x -4y +43 = 0. Therefore, the coefficients are:

a = 1 (coefficient of y²)

b = -16 (coefficient of x)

c = -4 (coefficient of y)

d = 43 (constant term)

Yes, so ordered quadruple is (1, -16, -4, 43).

Therefore, I think that's the answer.

But just to be thorough, let me check with a specific point. Let's take a point on the original circle, reflect it, and check if it satisfies the equation.

Take the point (5, 0) on the original circle. Reflecting over (4,1):

x' = 8 - 5 = 3

y' = 2 - 0 = 2

So the image point is (3, 2). Let's plug into the equation x² + y² -16x -4y +43:

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

Another point: (0,5). 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 = (73) -128 +12 +43 = 73 -128 is -55 +12 is -43 +43 = 0. Correct.

Another point: (0,0). Wait, (0,0) is on the original circle? Wait, 0² + 0² = 0 ≠ 25. So not on the circle. Let me pick (3,4). Original circle: 3² +4² = 9 +16 =25. Yes. Reflecting over (4,1):

x' = 8 - 3 = 5

y' = 2 - 4 = -2

So image point (5, -2). Plug into equation:

5² + (-2)^2 -16*5 -4*(-2) +43 = 25 +4 -80 +8 +43 = 29 -80 is -51 +8 is -43 +43 = 0. Correct.

Another test: (4,1) is the center of reflection. Let me take a point diametrically opposite to (4,1) on the original circle? Wait, original circle is centered at (0,0). The point (4,1) is not on the original circle. Let me take a point on the original circle, say (0,5). Wait, already did that.

Alternatively, take (sqrt(25 - y²), y). Maybe take (sqrt(25 - 0), 0) = (5,0), which we did.

Alternatively, take (sqrt(25 - 16), 4) = (3,4). Which we did.

So all test points work. Therefore, the equation is correct.

Therefore, the ordered quadruple is (1, -16, -4, 43).

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

To solve the problem, we are given a circle defined by the equation:

$$
x^2 + y^2 = 25
$$

We are to reflect **each point** on this circle over the point $(4, 1)$, and then determine the equation of the set of image points in the form:

$$
x^2 + a y^2 + b x + c y + d = 0
$$

---

### Step 1: Understanding the Reflection

Reflecting a point $(x, y)$ over a point $(h, k)$ results in a new point $(x', y')$ such that $(h, k)$ is the **midpoint** of the segment joining $(x, y)$ and $(x', y')$. Using the midpoint formula:

$$
\left( \frac{x + x'}{2}, \frac{y + y'}{2} \right) = (4, 1)
$$

Solving for $x'$ and $y'$:

$$
x' = 8 - x, \quad y' = 2 - y
$$

So, the image of a point $(x, y)$ on the original circle is $(8 - x, 2 - y)$.

---

### Step 2: Substituting into the Original Equation

Let the image point be $(X, Y)$. Then:

$$
X = 8 - x \Rightarrow x = 8 - X \\
Y = 2 - y \Rightarrow y = 2 - Y
$$

Substitute into the original circle equation:

$$
x^2 + y^2 = 25 \Rightarrow (8 - X)^2 + (2 - Y)^2 = 25
$$

---

### Step 3: Expanding the Equation

$$
(8 - X)^2 = 64 - 16X + X^2 \\
(2 - Y)^2 = 4 - 4Y + Y^2
$$

Adding both:

$$
64 - 16X + X^2 + 4 - 4Y + Y^2 = 25
$$

Simplify:

$$
X^2 + Y^2 - 16X - 4Y + 68 = 25
$$

Subtract 25 from both sides:

$$
X^2 + Y^2 - 16X - 4Y + 43 = 0
$$

---

### Step 4: Matching the Form

The problem asks for the equation in the form:

$$
x^2 + a y^2 + b x + c y + d = 0
$$

Comparing with our derived equation:

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

We identify the coefficients:

- $a = 1$ (coefficient of $y^2$)
- $b = -16$ (coefficient of $x$)
- $c = -4$ (coefficient of $y$)
- $d = 43$ (constant term)

---

### Final Answer

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

[Assistant]
Explanation:
Okay, so I need to figure out the equation of the set of image points after reflecting each point on the circle x² + y² = 25 over the point (4,1). The final equation is supposed to 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.

Reflecting a point P over a point Q results in a point P' such that Q is the midpoint of the segment PP'. So, if I have a point (x, y) on the original circle, its reflection (let's call it (x', y')) over (4,1) should satisfy the midpoint formula. That is, the midpoint between (x, y) and (x', y') is (4,1).

So, the midpoint formula says that:

( (x + x') / 2 , (y + y') / 2 ) = (4, 1)

Therefore, solving for x' and y', we get:

x + x' = 8 => x' = 8 - x

y + y' = 2 => y' = 2 - y

So, the reflection of any point (x, y) over (4,1) is (8 - x, 2 - y).

Now, since the original points lie on the circle x² + y² = 25, their reflections will satisfy some equation. Let me denote the reflected point as (X, Y). So, X = 8 - x and Y = 2 - y. Therefore, we can express x and y in terms of X and Y:

x = 8 - X

y = 2 - Y

Since (x, y) lies on the original circle, substituting these into the original equation:

(8 - X)² + (2 - Y)² = 25

Now, I need to expand this equation and rearrange it into the form x² + a y² + b x + c y + d = 0. Wait, but here the variables are X and Y, which are the coordinates of the image points. However, the problem says the set of image points satisfies the equation x² + a y² + b x + c y + d = 0. So, I need to write the equation in terms of X and Y, but since X and Y are just variables, we can just replace them with x and y for the equation. Let me check that.

So, expanding (8 - X)² + (2 - Y)² = 25:

First, expand (8 - X)²:

= 64 - 16X + X²

Then, expand (2 - Y)²:

= 4 - 4Y + Y²

Adding them together:

64 - 16X + X² + 4 - 4Y + Y² = 25

Combine like terms:

(64 + 4) + (-16X - 4Y) + (X² + Y²) = 25

Which is:

68 - 16X - 4Y + X² + Y² = 25

Now, subtract 25 from both sides to set the equation to zero:

X² + Y² - 16X - 4Y + 68 - 25 = 0

Calculating 68 - 25:

68 - 25 = 43

So, the equation becomes:

X² + Y² - 16X - 4Y + 43 = 0

But the problem states that the equation is in the form x² + a y² + b x + c y + d = 0. Comparing this with our equation, we can see that:

x² + y² - 16x - 4y + 43 = 0

Therefore, the coefficients are:

a = 1 (since the coefficient of y² is 1)

b = -16

c = -4

d = 43

Wait, but the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". So, in our case, the equation is x² + y² -16x -4y +43 = 0, so comparing term by term:

x² + (1)y² + (-16)x + (-4)y + 43 = 0. Therefore, the ordered quadruple is (a, b, c, d) = (1, -16, -4, 43).

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

First, reflection over a point: yes, midpoint formula. So if original point is (x, y), then reflection is (8 - x, 2 - y). Then substituting into original equation.

Original equation: x² + y² = 25. So replacing x with 8 - X and y with 2 - Y. Then expanding:

(8 - X)^2 + (2 - Y)^2 = 25.

Expanding:

(64 - 16X + X²) + (4 - 4Y + Y²) = 25.

Adding 64 + 4 = 68. Then -16X -4Y. Then X² + Y².

So 68 -16X -4Y + X² + Y² = 25.

Subtract 25: X² + Y² -16X -4Y + 43 = 0.

Yes, that seems correct.

But let me check if there's a possibility that the problem expects the equation in terms of the original variables? Wait, no. The image points are (X, Y), so the equation they satisfy is in terms of X and Y. But since the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0", so they just use x and y as variables for the image points. So replacing X and Y with x and y, the equation is x² + y² -16x -4y +43 = 0. Therefore, the coefficients are a=1, b=-16, c=-4, d=43.

But let me check if there's a different approach. Maybe I can think about the reflection of the circle. Reflecting a circle over a point... Well, reflecting a circle over a point results in another circle. The original circle has center at (0,0) and radius 5. Reflecting over the point (4,1) would result in a circle whose center is the reflection of the original center over (4,1).

Let me try this approach. The original center is (0,0). Reflecting (0,0) over (4,1) gives a point (8, 2), since midpoint between (0,0) and (8,2) is (4,1). Therefore, the reflected circle has center (8,2) and same radius 5. Therefore, the equation is (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 + 43 = 0.

Same result as before! So that confirms that the equation is indeed x² + y² -16x -4y +43 = 0. Therefore, the ordered quadruple is (1, -16, -4, 43).

But let me check once more if the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". So, in our case, the equation is x² + y² -16x -4y +43 = 0. Therefore, comparing term by term:

x² + (1)y² + (-16)x + (-4)y + 43 = 0. Therefore, a=1, b=-16, c=-4, d=43.

So the ordered quadruple is (1, -16, -4, 43).

But let me check if there's any possible mistake in the reflection. For example, if I had messed up the reflection formula. Let me think again. If you reflect a point (x, y) over the point (h, k), then the reflection point (x', y') is such that (h, k) is the midpoint. Therefore:

h = (x + x')/2 => x' = 2h - x

Similarly, y' = 2k - y.

So in this case, h = 4, k = 1. Therefore, x' = 8 - x, y' = 2 - y. Which is what I had before. So that's correct.

Therefore, substituting into original equation gives the equation of the reflected circle.

Alternatively, if I think of the reflected circle as having center (8, 2) and radius 5, then equation is (x - 8)^2 + (y - 2)^2 = 25. Expanding that gives x² -16x +64 + y² -4y +4 = 25 => x² + y² -16x -4y + 68 -25 = 0 => x² + y² -16x -4y +43 = 0.

Same result. So that's correct.

Therefore, the answer is (a, b, c, d) = (1, -16, -4, 43).

But let me check if the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d = 0". So, in our case, the equation is x² + y² -16x -4y +43 = 0. Therefore, the coefficients are:

a = 1 (coefficient of y²)

b = -16 (coefficient of x)

c = -4 (coefficient of y)

d = 43 (constant term)

Yes, so ordered quadruple is (1, -16, -4, 43).

Therefore, I think that's the answer.

But just to be thorough, let me check with a specific point. Let's take a point on the original circle, reflect it, and check if it satisfies the equation.

Take the point (5, 0) on the original circle. Reflecting over (4,1):

x' = 8 - 5 = 3

y' = 2 - 0 = 2

So the image point is (3, 2). Let's plug into the equation x² + y² -16x -4y +43:

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

Another point: (0,5). 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 = (73) -128 +12 +43 = 73 -128 is -55 +12 is -43 +43 = 0. Correct.

Another point: (0,0). Wait, (0,0) is on the original circle? Wait, 0² + 0² = 0 ≠ 25. So not on the circle. Let me pick (3,4). Original circle: 3² +4² = 9 +16 =25. Yes. Reflecting over (4,1):

x' = 8 - 3 = 5

y' = 2 - 4 = -2

So image point (5, -2). Plug into equation:

5² + (-2)^2 -16*5 -4*(-2) +43 = 25 +4 -80 +8 +43 = 29 -80 is -51 +8 is -43 +43 = 0. Correct.

Another test: (4,1) is the center of reflection. Let me take a point diametrically opposite to (4,1) on the original circle? Wait, original circle is centered at (0,0). The point (4,1) is not on the original circle. Let me take a point on the original circle, say (0,5). Wait, already did that.

Alternatively, take (sqrt(25 - y²), y). Maybe take (sqrt(25 - 0), 0) = (5,0), which we did.

Alternatively, take (sqrt(25 - 16), 4) = (3,4). Which we did.

So all test points work. Therefore, the equation is correct.

Therefore, the ordered quadruple is (1, -16, -4, 43).

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

To solve the problem, we are given a circle defined by the equation:

$$
x^2 + y^2 = 25
$$

We are to reflect **each point** on this circle over the point $(4, 1)$, and then determine the equation of the set of image points in the form:

$$
x^2 + a y^2 + b x + c y + d = 0
$$

---

### Step 1: Understanding the Reflection

Reflecting a point $(x, y)$ over a point $(h, k)$ results in a new point $(x', y')$ such that $(h, k)$ is the **midpoint** of the segment joining $(x, y)$ and $(x', y')$. Using the midpoint formula:

$$
\left( \frac{x + x'}{2}, \frac{y + y'}{2} \right) = (4, 1)
$$

Solving for $x'$ and $y'$:

$$
x' = 8 - x, \quad y' = 2 - y
$$

So, the image of a point $(x, y)$ on the original circle is $(8 - x, 2 - y)$.

---

### Step 2: Substituting into the Original Equation

Let the image point be $(X, Y)$. Then:

$$
X = 8 - x \Rightarrow x = 8 - X \\
Y = 2 - y \Rightarrow y = 2 - Y
$$

Substitute into the original circle equation:

$$
x^2 + y^2 = 25 \Rightarrow (8 - X)^2 + (2 - Y)^2 = 25
$$

---

### Step 3: Expanding the Equation

$$
(8 - X)^2 = 64 - 16X + X^2 \\
(2 - Y)^2 = 4 - 4Y + Y^2
$$

Adding both:

$$
64 - 16X + X^2 + 4 - 4Y + Y^2 = 25
$$

Simplify:

$$
X^2 + Y^2 - 16X - 4Y + 68 = 25
$$

Subtract 25 from both sides:

$$
X^2 + Y^2 - 16X - 4Y + 43 = 0
$$

---

### Step 4: Matching the Form

The problem asks for the equation in the form:

$$
x^2 + a y^2 + b x + c y + d = 0
$$

Comparing with our derived equation:

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

We identify the coefficients:

- $a = 1$ (coefficient of $y^2$)
- $b = -16$ (coefficient of $x$)
- $c = -4$ (coefficient of $y$)
- $d = 43$ (constant term)

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### 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, $(a,b,c,d) = \boxed{(1,-16,-4,43)}.$