Explanation:
Okay, so I need to find the degree measure of the smallest angle in triangle PQS. The diagram shows four circles of radius 1 with centers P, Q, R, and S tangent to each other and to the sides of triangle ABC. Let me try to visualize this. There's a triangle ABC with three circles near the corners and one circle somewhere inside? Wait, the Asymptote code shows coordinates for points R, Q, S, and P. Let me check the coordinates given in the Asymptote code:

R is at (0,0), Q is at (-2,0), S is at (2,0), and P is at (1, 1.732). Then points A, B, C are given with coordinates A=(1.366,3.098), B=(-5.73,-1), C=(3.732,-1).

Wait, so maybe triangle ABC is a large triangle with base BC, and points Q, R, S are on the base? Let me think. The centers of the circles are P, Q, R, S. Each circle has radius 1, and they are tangent to each other and to the sides of triangle ABC.

So, since Q, R, S are on the base BC? Let me check their coordinates. R is at (0,0), Q is at (-2,0), S is at (2,0). So they are on the x-axis. Then P is at (1, 1.732). 1.732 is approximately √3, which is about 1.732. So P is above the x-axis.

So maybe triangle ABC has its base BC on the x-axis, with points Q, R, S on BC? But Q is at (-2,0), R at (0,0), S at (2,0). So BC is from B to C, which in the Asymptote code is from (-5.73,-1) to (3.732,-1). Wait, but R is at (0,0), which is above the base BC? Wait, no, BC is at y=-1? Wait, the coordinates of B and C are (-5.73,-1) and (3.732,-1). So BC is a horizontal line at y=-1. Then R is at (0,0), which is above BC. So maybe the triangle ABC is a large triangle with base BC at y=-1, and vertex A somewhere above.

But then the circles are tangent to the sides of ABC. So each circle is tangent to two sides of the triangle? For example, circle Q is tangent to sides AB and BC? Similarly for the others?

But since there are four circles, maybe two circles near the base BC, one near the left side AB, and one near the right side AC? Wait, but the centers are P, Q, R, S. Let me think.

Alternatively, maybe the triangle ABC is such that each side has a circle tangent to it. But there are four circles. Maybe two circles on the base BC, one on AB, and one on AC? But the Asymptote code shows that R is at (0,0), Q at (-2,0), S at (2,0). So maybe Q and S are on the base BC? But BC is at y=-1. Wait, but R is at (0,0), which is above BC. Maybe the circles are tangent to the sides of ABC and to each other.

Alternatively, maybe the triangle ABC is an equilateral triangle? But the coordinates given for A are (1.366, 3.098). Let me check if that's an equilateral triangle. If ABC is equilateral, then all sides are equal. Let me compute the distance between B and C. B is at (-5.73, -1), C is at (3.732, -1). The distance between them is 3.732 - (-5.73) = 9.462 units? Wait, but the coordinates are approximate. Maybe they are using specific values.

Alternatively, maybe the triangle ABC is a 30-60-90 triangle? Or some other triangle. But maybe I don't need to figure out the exact coordinates of ABC. Maybe I can use the positions of the centers P, Q, R, S.

Since all circles have radius 1, and they are tangent to each other. So the distance between centers of tangent circles should be 2 units. Let me check the distances between Q, R, S, and P.

First, Q is at (-2,0), R is at (0,0). The distance between Q and R is 2 units. Since they are both radius 1, so they are tangent. Similarly, R is at (0,0), S is at (2,0), distance is 2 units. So QR and RS are each 2 units. So Q, R, S are colinear on the x-axis, each 2 units apart.

Now, what about the other circles? Let's check the distance between P and Q. P is at (1, √3) approximately. Let me compute the distance between P(1, √3) and Q(-2, 0). The distance squared is (1 - (-2))² + (√3 - 0)² = (3)² + (√3)² = 9 + 3 = 12. So the distance is √12 = 2√3 ≈ 3.464. But since the circles have radius 1, if they are tangent, the distance between centers should be 2. But 2√3 is more than 2, so they are not tangent. Wait, but the problem says the circles are tangent to one another. So maybe my assumption about the coordinates is wrong? Or maybe the diagram is not to scale?

Wait, the Asymptote code is just for drawing, but maybe the coordinates are approximate. Let me check the coordinates again. The Asymptote code defines R=(0,0), Q=(-2,0), S=(2,0), P=(1,1.732). So 1.732 is approximately √3. So P is at (1, √3). Then the distance between P and Q is sqrt[(1 - (-2))² + (√3 - 0)^2] = sqrt[9 + 3] = sqrt[12] = 2*sqrt(3). Which is about 3.464. But if the circles are tangent, the distance between centers should be 2. So that suggests that maybe the diagram is not accurate? Or maybe the circles are tangent to the sides of the triangle, not necessarily to each other? Wait, the problem says "four circles of radius 1 with centers P, Q, R, and S are tangent to one another and to the sides of triangle ABC". So they are tangent to each other and to the sides.

Therefore, the distance between centers of tangent circles should be 2. So if P is tangent to Q, then the distance between P and Q should be 2. But according to coordinates, it's 2√3. So there's a contradiction here. Therefore, maybe the coordinates given in the Asymptote code are not exact? Or maybe I need to figure out the actual coordinates based on the tangency conditions?

Alternatively, maybe the Asymptote code is illustrative, but the actual problem is more general. Maybe I need to solve it without relying on the coordinates?

Alternatively, maybe the triangle ABC is such that the centers P, Q, R, S are located in a specific way. Let me think.

Since there are four circles tangent to each other and to the sides of triangle ABC. This seems similar to the problem of circles tangent to two sides of a triangle and to another circle. Maybe it's related to the Soddy circles or something like that.

Alternatively, since there are four circles, maybe two circles are tangent to the base BC and the other sides, and the other two circles are tangent to the other sides?

Alternatively, maybe triangle ABC is such that there are three circles tangent to two sides and the incircle? But there are four circles here.

Alternatively, maybe the triangle ABC is divided into smaller regions by the circles.

Alternatively, since the problem is about triangle PQS, maybe I need to find coordinates of P, Q, S based on the tangency conditions.

Let me try to think step by step.

First, let me assume that triangle ABC is such that there are four circles of radius 1 tangent to each other and to the sides. Let me try to figure out the positions of P, Q, R, S.

Let me consider that Q, R, S are on the base BC. Since they are tangent to each other and to the sides. If Q and R are on BC, then they are tangent to BC. Similarly, S is also on BC. Wait, but if Q, R, S are all on BC, then they would be tangent to BC, but also tangent to each other. So the distance between Q and R is 2 units (since each has radius 1), same with R and S. So if Q, R, S are on BC, then BC must be long enough to accommodate them.

But if Q is at distance 2 from R, and R is at distance 2 from S, then the total length of BC would be QR + RS = 2 + 2 = 4 units? But then there's also the circle at S, which is tangent to BC and maybe to another side?

Alternatively, maybe there are two circles on the base BC, each tangent to BC and to the other circle, and then another circle tangent to AB and AC? But there are four circles.

Alternatively, maybe there are two circles on each of two sides?

Alternatively, maybe the triangle ABC is such that each side has two circles tangent to it?

This is getting confusing. Maybe I need to think about the configuration.

In some problems, there are circles tangent to two sides of a triangle and to another circle. For example, the mixtilinear incircles. But here, there are four circles.

Alternatively, maybe the triangle ABC is such that there are three circles tangent to two sides and the incircle, and then the incircle itself? But that would be four circles. However, the problem says the circles are tangent to one another and to the sides.

Alternatively, maybe the triangle ABC is divided into smaller triangles by the circles, and the centers form triangle PQS.

Alternatively, since the Asymptote code gives coordinates, maybe I can use those coordinates to calculate the angles? But earlier, the distance between P and Q was 2√3, which is inconsistent with them being tangent. However, maybe the diagram is not to scale?

Wait, the problem says "four circles of radius 1 with centers P, Q, R, and S are tangent to one another and to the sides of triangle ABC". So if they are tangent to each other, then the distance between centers should be 2. So if in the Asymptote code, the distance between P and Q is 2√3, which is more than 2, that suggests that maybe the diagram is not accurate? Or maybe the coordinates are approximate?

Alternatively, maybe the Asymptote code is using specific coordinates for illustration, but the actual problem requires a different approach.

Alternatively, maybe I can use coordinates to find the positions of P, Q, R, S based on the tangency conditions. Let me try that.

Let me assume that triangle ABC is such that there are four circles of radius 1 tangent to each other and to the sides. Let me try to model this.

Let me consider that Q, R, S are on the base BC. Let me assume that BC is horizontal. Let me suppose that Q is tangent to BC and to AB, R is tangent to BC and to AC, and S is tangent to BC and to... Wait, but there are four circles. Maybe there is another circle tangent to AB and AC?

Alternatively, maybe there are two circles on the base BC, each tangent to BC and to the other circle, and then two circles tangent to AB and AC?

Alternatively, maybe the configuration is similar to the problem of circles tangent to two sides and to another circle.

Alternatively, since there are four circles, maybe two circles are tangent to the base BC and to the other two sides, and the other two circles are tangent to the other sides?

This is getting too vague. Maybe I need to think about the triangle ABC.

Alternatively, since the Asymptote code gives coordinates for A, B, C, maybe I can use those coordinates to find the positions of P, Q, R, S?

Wait, the Asymptote code defines:

R=(0,0);

Q=(-2,0);

S=(2,0);

P=(1,1.732);

B=(-5.73,-1);

C=(3.732,-1);

A=(1.366,3.098);

So, if I can use these coordinates, even though the distances between centers may not be exactly 2, maybe I can compute the angles of triangle PQS?

But the problem says the circles are tangent to each other and to the sides. So maybe the coordinates are approximate?

Alternatively, maybe the coordinates are exact? Let me check.

First, let me check the coordinates of P=(1, 1.732). 1.732 is approximately √3, which is about 1.732. So maybe P is at (1, √3). Similarly, Q is at (-2, 0), R at (0,0), S at (2,0).

So if I take these coordinates as exact, then let me compute the distances between P, Q, R, S.

Distance between Q(-2, 0) and R(0,0) is 2 units. Good, since they are tangent.

Distance between R(0,0) and S(2,0) is 2 units. Good.

Distance between Q(-2,0) and S(2,0) is 4 units.

Distance between P(1, √3) and Q(-2,0): sqrt[(1 - (-2))² + (√3 - 0)^2] = sqrt[9 + 3] = sqrt[12] = 2*sqrt(3) ≈ 3.464.

Distance between P(1, √3) and R(0,0): sqrt[(1 - 0)^2 + (√3 - 0)^2] = sqrt[1 + 3] = sqrt[4] = 2. So that's good! So P is tangent to R.

Similarly, distance between P(1, √3) and S(2,0): sqrt[(1 - 2)^2 + (√3 - 0)^2] = sqrt[1 + 3] = 2. So P is tangent to S as well.

So P is tangent to Q? Wait, no. Wait, P is tangent to R and S, but not to Q? Wait, the distance between P and Q is 2√3, which is more than 2, so they are not tangent. But the problem says all four circles are tangent to one another. Wait, so maybe there's a mistake here?

Wait, the problem says "four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of triangle ABC". So all four circles are tangent to each other. So each pair of circles should be tangent. But according to the coordinates, P is tangent to R and S, but not to Q. Similarly, Q is tangent to R, R is tangent to S, but Q is not tangent to S? Wait, distance between Q(-2,0) and S(2,0) is 4 units, which is more than 2, so they are not tangent.

So there's a contradiction here. Therefore, the Asymptote code's coordinates must be illustrative, not exact. Therefore, I can't rely on them for exact distances.

Therefore, I need to figure out the positions of P, Q, R, S based on the tangency conditions.

Let me try again. Let me assume that triangle ABC is such that there are four circles of radius 1 tangent to each other and to the sides. Let me think about the configuration.

In such problems, often the centers of the circles lie along the angle bisectors of the triangle. However, with four circles, it's more complex.

Alternatively, maybe the triangle ABC is such that there are two circles tangent to the base BC and to the other sides, and two circles tangent to the other sides?

Alternatively, since there are four circles, maybe two circles are tangent to the base BC and to the other two sides, and the other two circles are tangent to the other sides?

Alternatively, maybe the triangle ABC is divided into smaller triangles by the circles, and the centers form a smaller triangle.

Alternatively, since the problem is about triangle PQS, maybe I can find coordinates for P, Q, S based on the tangency conditions.

Let me consider that Q, R, S are on the base BC. Let me assume that BC is horizontal. Let me suppose that Q and S are on BC, each tangent to BC and to the other circles. Also, R is between Q and S? Wait, but if there are three circles on BC, each tangent to BC and to each other, then the distance between Q and R is 2, and between R and S is 2, so total length of BC would be QR + RS = 4. But then there's also the circle P, which is tangent to Q, R, and S? Wait, but if P is tangent to Q, R, and S, then it's the circumcircle? But all circles have radius 1.

Alternatively, maybe P is tangent to Q and S, and also tangent to another side?

Alternatively, maybe the triangle ABC is such that there are two circles on the base BC, each tangent to BC and to the other circle, and then two circles tangent to the other sides.

Alternatively, maybe the triangle ABC is such that there are three circles tangent to two sides and the incircle, and the incircle itself? But that would be four circles.

Alternatively, maybe the triangle ABC is equilateral? If ABC is equilateral, then maybe the circles are arranged symmetrically.

Let me try assuming ABC is equilateral. Let me suppose ABC is equilateral with side length 'a'. Then, the inradius is (a√3)/6. If there are circles of radius 1 tangent to the sides, then maybe the inradius is related. However, with four circles, it's more complex.

Alternatively, maybe the triangle ABC is a right triangle?

Alternatively, since the Asymptote code has coordinates for A, B, C, maybe I can use those coordinates to find the positions of P, Q, R, S?

Wait, let me check the coordinates again.

In the Asymptote code:

A=(1.366,3.098)

B=(-5.73,-1)

C=(3.732,-1)

So, coordinates of B and C are at y=-1, so BC is horizontal. The coordinates of A are approximately (1.366, 3.098). Let me check if ABC is a specific triangle.

Let me compute the lengths of AB, BC, and AC.

First, BC: distance between B(-5.73, -1) and C(3.732, -1) is 3.732 - (-5.73) = 9.462 units.

AB: distance between A(1.366, 3.098) and B(-5.73, -1). Let me compute:

Δx = 1.366 - (-5.73) = 7.096

Δy = 3.098 - (-1) = 4.098

Distance AB = sqrt(7.096² + 4.098²). Let me approximate:

7.096² ≈ 50.35

4.098² ≈ 16.79

Total ≈ 67.14, so sqrt(67.14) ≈ 8.194

Similarly, AC: distance between A(1.366, 3.098) and C(3.732, -1):

Δx = 3.732 - 1.366 = 2.366

Δy = -1 - 3.098 = -4.098

Distance AC = sqrt(2.366² + 4.098²) ≈ sqrt(5.598 + 16.79) ≈ sqrt(22.388) ≈ 4.732

So sides are approximately BC ≈ 9.46, AB ≈ 8.19, AC ≈ 4.73. So triangle ABC is scalene.

Now, the centers of the circles are P, Q, R, S. Let me check their coordinates:

R=(0,0), Q=(-2,0), S=(2,0), P=(1,1.732).

So, if I consider the triangle ABC with coordinates as given, then the circles are located as follows:

- Q is at (-2, 0), which is inside the triangle? Let me check if that's possible.

Point B is at (-5.73, -1), so moving from B to C, which is at (3.732, -1). The base BC is from x=-5.73 to x=3.732 at y=-1. The point Q is at (-2, 0), which is above the base BC. Similarly, R is at (0,0), also above BC, and S is at (2,0), also above BC. P is at (1, 1.732), which is above R and S.

So maybe these circles are located near the sides of the triangle. For example, circle Q is tangent to AB and BC, circle R is tangent to BC and... maybe another side?

Alternatively, since the circles are tangent to the sides of ABC, each circle is tangent to two sides. For example, circle Q is tangent to AB and BC, circle R is tangent to BC and... maybe AC? Wait, but there are four circles.

Alternatively, maybe two circles are tangent to BC and the other sides, and two circles are tangent to AB and AC?

Alternatively, since there are four circles, maybe each circle is tangent to two sides and to another circle.

Alternatively, since the problem says they are tangent to one another and to the sides, so each circle is tangent to two sides of the triangle and to two other circles.

But with four circles, it's complex.

Alternatively, maybe the triangle ABC is such that there are two circles tangent to AB and BC, two circles tangent to BC and AC, and one circle tangent to AB and AC? But that would be five circles.

Alternatively, maybe the four circles are arranged such that two are on the base BC, each tangent to BC and to the other circle, and the other two are tangent to AB and AC.

But how would they be tangent to each other?

Alternatively, maybe the centers P, Q, R, S form a specific quadrilateral.

But since the problem is about triangle PQS, maybe I can find coordinates of P, Q, S based on the tangency conditions.

Let me try to think about circle Q. If it's tangent to BC and AB, then its center Q is located at a distance of 1 from BC and from AB. Similarly for other circles.

Given that BC is the base at y=-1, then the distance from Q to BC is 1. Since BC is at y=-1, the y-coordinate of Q should be -1 + 1 = 0. Which matches the coordinates of Q at (-2, 0). Similarly, R is at (0,0), so distance from BC is 1. Similarly for S at (2,0).

So Q, R, S are all at y=0, which is 1 unit above BC.

Now, circle Q is also tangent to AB. So the distance from Q to AB is 1. Similarly for circle R and AC?

Let me find the equation of AB to check.

Point A is at (1.366, 3.098), point B is at (-5.73, -1). Let me compute the equation of line AB.

First, compute the slope of AB:

m = (y_A - y_B)/(x_A - x_B) = (3.098 - (-1))/(1.366 - (-5.73)) = (4.098)/(7.096) ≈ 0.577.

Which is approximately 1/√3 ≈ 0.577. So the slope is approximately 1/√3.

Therefore, the equation of AB can be written as y - y_B = m(x - x_B).

So y - (-1) = (1/√3)(x - (-5.73)) => y + 1 = (1/√3)(x + 5.73).

But since the distance from Q(-2, 0) to AB is 1, let me verify that.

The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²).

First, let me write the equation of AB in standard form.

Starting from y + 1 = (1/√3)(x + 5.73). Multiply both sides by √3:

√3(y + 1) = x + 5.73

Rearranged: x - √3 y + (5.73 - √3) = 0.

Wait, let me check:

Starting from y + 1 = (1/√3)(x + 5.73)

Multiply both sides by √3:

√3 y + √3 = x + 5.73

Bring all terms to left:

x - √3 y + (5.73 - √3) = 0

So the standard form is x - √3 y + (5.73 - √3) = 0.

Now, the distance from Q(-2, 0) to this line is |(-2) - √3*(0) + (5.73 - √3)| / sqrt(1 + (√3)^2) = | -2 + 5.73 - √3 | / sqrt(1 + 3) = |3.73 - √3| / 2.

Compute 3.73 - √3 ≈ 3.73 - 1.732 ≈ 1.998 ≈ 2. So the distance is approximately 2 / 2 = 1. Which matches the required distance of 1.

Therefore, the distance from Q to AB is indeed 1, which means circle Q is tangent to AB and BC. Similarly, circle S is tangent to AC and BC.

Now, what about circle R? It is at (0,0), which is 1 unit above BC. Let me check if it is tangent to AC.

First, find the equation of AC.

Point A is (1.366, 3.098), point C is (3.732, -1).

Slope of AC: m = (-1 - 3.098)/(3.732 - 1.366) = (-4.098)/2.366 ≈ -1.732. Which is approximately -√3.

So the equation of AC is y - y_C = m(x - x_C).

So y - (-1) = -√3(x - 3.732) => y + 1 = -√3 x + 3.732√3.

Rearranged: √3 x + y + (1 - 3.732√3) = 0.

Now, compute the distance from R(0,0) to this line.

Distance = |√3*0 + 1*0 + (1 - 3.732√3)| / sqrt( (√3)^2 + 1^2 ) = |1 - 3.732√3| / 2.

Compute 3.732√3 ≈ 3.732 * 1.732 ≈ 6.464. So 1 - 6.464 ≈ -5.464. Absolute value is 5.464. Divided by 2 is ≈ 2.732. Which is not equal to 1.

Wait, that's a problem. The distance from R to AC is approximately 2.732, but it should be 1 since the circle is tangent to AC. Therefore, there's inconsistency here.

But according to the Asymptote code, R is at (0,0), which is supposed to be the center of a circle tangent to the sides. So maybe my calculation is wrong?

Wait, let me check the equation of AC again.

Point A is (1.366, 3.098), point C is (3.732, -1).

Let me compute the slope again:

m = (y_C - y_A)/(x_C - x_A) = (-1 - 3.098)/(3.732 - 1.366) = (-4.098)/2.366 ≈ -1.732. Which is -√3.

So the slope is -√3.

Equation of AC: y - y_A = m(x - x_A).

So y - 3.098 = -√3(x - 1.366).

Let me rearrange this:

y = -√3 x + √3 * 1.366 + 3.098.

Compute √3 * 1.366 ≈ 1.732 * 1.366 ≈ 2.366.

So y ≈ -√3 x + 2.366 + 3.098 ≈ -√3 x + 5.464.

Therefore, the equation of AC is y + √3 x - 5.464 = 0.

So standard form: √3 x + y - 5.464 = 0.

Now, distance from R(0,0) to this line is |√3*0 + 1*0 - 5.464| / sqrt( (√3)^2 + 1^2 ) = | -5.464 | / 2 ≈ 5.464 / 2 ≈ 2.732.

Which is not 1. So this suggests that circle R is not tangent to AC. But according to the problem statement, all circles are tangent to the sides. Therefore, there must be an error in my assumption.

Alternatively, maybe circle R is tangent to AB and BC? But then it would be similar to Q. But Q is already tangent to AB and BC.

Alternatively, maybe circle R is tangent to BC and another side?

Wait, if R is at (0,0), which is 1 unit above BC, and if it's tangent to another side, say AB or AC. But we saw that distance to AB is... Let me check distance from R(0,0) to AB.

Earlier, the equation of AB was x - √3 y + (5.73 - √3) = 0.

Wait, earlier I had:

From AB: y + 1 = (1/√3)(x + 5.73).

Which rearranged to x - √3 y + (5.73 - √3) = 0.

So distance from R(0,0) to AB is |0 - √3*0 + 5.73 - √3| / 2 = |5.73 - √3| / 2 ≈ |5.73 - 1.732| / 2 ≈ 3.998 / 2 ≈ 1.999 ≈ 2.

Which is not 1. So circle R is not tangent to AB.

So this suggests that the Asymptote code's coordinates are not consistent with the problem's conditions. Therefore, I need to find another approach.

Let me try to think of the configuration again.

Assume that there are four circles of radius 1, tangent to each other and to the sides of triangle ABC. Let me consider that two of the circles are on the base BC, each tangent to BC and to each other, and the other two circles are tangent to the other sides.

Let me denote the two circles on BC as Q and R, with centers at distance 2 units apart. Let me assume that the base BC is horizontal, and the circles Q and R are located above BC, each tangent to BC. Then, the centers Q and R are at height 1 above BC.

Now, there are two more circles, P and S, which are tangent to the other sides. Let me assume that circle P is tangent to AB and AC, and circle S is tangent to AB and AC? No, that would be two circles.

Alternatively, maybe circle P is tangent to AB and BC, and circle S is tangent to AC and BC? But then we already have Q and R on BC.

Wait, if there are four circles, maybe two on BC (Q and R), one on AB (P), and one on AC (S)?

But then how are they tangent to each other?

Alternatively, maybe circle P is tangent to AB, AC, and Q, and circle S is tangent to AB, AC, and R?

But this is getting too vague.

Alternatively, since the problem mentions triangle PQS, maybe I can find the coordinates of P, Q, S based on the tangency conditions.

Let me consider that Q and R are on BC, each at distance 1 from BC, and distance 2 apart. Let me assume BC is horizontal, and let me place coordinate system such that BC is on the x-axis, with B at (0,0), C at (c, 0), and Q at (q, 1), R at (r, 1). Since they are tangent to each other, the distance between Q and R is 2, so |r - q| = 2.

But also, they are tangent to BC, so their y-coordinate is 1.

Now, there are other circles. Let me assume that there is a circle P tangent to AB, AC, and Q. Similarly, a circle S tangent to AB, AC, and R?

Alternatively, maybe there is a circle tangent to AB, BC, and Q, and another circle tangent to AC, BC, and R?

But then there would be four circles: Q, R, P, S.

Let me try this.

Let me assume that circle Q is tangent to AB, BC, and circle R. Similarly, circle R is tangent to AC, BC, and circle Q. Wait, but then there would be two circles on BC, each tangent to one side and the base.

Alternatively, maybe circle Q is tangent to AB, BC, and circle P, and circle R is tangent to AC, BC, and circle P?

This is getting too complex. Maybe I need to use coordinate geometry.

Let me try to set up coordinates. Let me assume that BC is on the x-axis, with B at (0,0), C at (c, 0). Let me assume that circle Q is tangent to BC and AB, and circle R is tangent to BC and AC. Also, circles Q and R are tangent to each other.

Let me denote the coordinates of Q as (x1, 1), since it's 1 unit above BC. Similarly, R is at (x2, 1). The distance between Q and R is 2, so sqrt[(x2 - x1)^2 + (1 - 1)^2] = |x2 - x1| = 2. So x2 = x1 + 2 or x1 = x2 + 2.

Now, circle Q is tangent to AB. Let me find the equation of AB. Let me assume that A is at some point (a, b).

But since circle Q is tangent to AB and BC, and has center at (x1, 1), the distance from Q to AB is 1. Similarly, the distance from Q to BC is 1, which is already satisfied since y-coordinate is 1.

Similarly for circle R.

Let me find the coordinates of A. Since circle Q is tangent to AB, and circle R is tangent to AC, and the triangle ABC has AB and AC as sides.

This seems too abstract. Maybe I need to relate the coordinates of A with the positions of Q and R.

Alternatively, since there are four circles, maybe the triangle ABC is such that it has an incircle and three excircles? No, that's not helpful.

Alternatively, since there are four circles, maybe it's a special triangle where the circles are arranged symmetrically.

Wait, going back to the Asymptote code, even though the distances between P and Q are not 2, maybe I can use the coordinates to calculate the angles of triangle PQS?

Given that in the Asymptote code, P is at (1, √3), Q is at (-2, 0), S is at (2, 0). Let me compute the lengths of the sides of triangle PQS.

First, PQ: distance between P(1, √3) and Q(-2, 0):

sqrt[(1 - (-2))² + (√3 - 0)²] = sqrt[9 + 3] = sqrt[12] = 2*sqrt(3) ≈ 3.464

QS: distance between Q(-2, 0) and S(2, 0): sqrt[(2 - (-2))² + (0 - 0)^2] = sqrt[16] = 4

PS: distance between P(1, √3) and S(2, 0):

sqrt[(1 - 2)^2 + (√3 - 0)^2] = sqrt[1 + 3] = sqrt[4] = 2

So triangle PQS has sides of length 2, 2√3, and 4.

Wait, but in a triangle, the sum of any two sides must be greater than the third. Let's check:

2 + 2√3 ≈ 2 + 3.464 = 5.464 > 4

2 + 4 = 6 > 2√3 ≈ 3.464

2√3 + 4 ≈ 3.464 + 4 = 7.464 > 2

So it's a valid triangle.

Now, to find the smallest angle in triangle PQS. The smallest angle is opposite the shortest side. The sides are 2, 2√3, and 4. The shortest side is 2, which is PS. Therefore, the smallest angle is the angle opposite PS, which is angle Q.

Wait, no. Wait, in triangle PQS, the sides are:

PQ = 2√3 ≈ 3.464

QS = 4

PS = 2

So the sides are PS = 2, PQ = 2√3, QS = 4.

Therefore, the smallest side is PS = 2, so the angle opposite to it is angle Q. Wait, no. In triangle PQS, the side opposite angle P is QS = 4, opposite angle Q is PS = 2, and opposite angle S is PQ = 2√3.

Therefore, the smallest angle is angle Q, which is opposite the shortest side PS = 2.

Wait, but let me confirm. In triangle PQS, the angles are at P, Q, S.

Side opposite angle P is QS = 4

Side opposite angle Q is PS = 2

Side opposite angle S is PQ = 2√3

Therefore, the smallest angle is angle Q.

But the question asks for the degree measure of the smallest angle in triangle PQS. So I need to find angle Q.

Alternatively, maybe I can use the Law of Cosines to find the angles.

Let me compute angle at Q.

In triangle PQS, angle at Q is between sides QP and QS.

Wait, sides adjacent to angle Q are QP and QS. Wait, no. Wait, angle at Q is between sides QP and QR? Wait, no, triangle PQS has vertices P, Q, S. So angle at Q is between sides QP and QS.

Wait, the sides adjacent to angle Q are QP and QS. Wait, but QS is the side from Q to S, which is length 4, and QP is from Q to P, which is length 2√3. The side opposite angle Q is PS, which is length 2.

So using the Law of Cosines on angle Q:

PS² = QP² + QS² - 2 * QP * QS * cos(angle Q)

So:

2² = (2√3)² + 4² - 2 * (2√3) * 4 * cos(angle Q)

Compute:

4 = 12 + 16 - 16√3 * cos(angle Q)

Simplify:

4 = 28 - 16√3 * cos(angle Q)

Rearranged:

16√3 * cos(angle Q) = 28 - 4 = 24

Therefore:

cos(angle Q) = 24 / (16√3) = (3) / (2√3) = √3 / 2

Therefore, angle Q is arccos(√3 / 2) = 30 degrees.

So the smallest angle in triangle PQS is 30 degrees.

But wait, let me check if this is correct.

Alternatively, since the coordinates are given, maybe I can compute the angle using coordinates.

Coordinates:

P(1, √3), Q(-2, 0), S(2, 0)

Vector QP = P - Q = (1 - (-2), √3 - 0) = (3, √3)

Vector QS = S - Q = (2 - (-2), 0 - 0) = (4, 0)

The angle at Q is between vectors QP and QS.

The formula for the angle between two vectors u and v is:

cosθ = (u • v) / (|u| |v|)

Compute the dot product of QP and QS:

(3)(4) + (√3)(0) = 12 + 0 = 12

|QP| = sqrt(3² + (√3)^2) = sqrt(9 + 3) = sqrt(12) = 2√3

|QS| = sqrt(4² + 0²) = 4

Therefore, cosθ = 12 / (2√3 * 4) = 12 / (8√3) = 3 / (2√3) = √3 / 2

Which gives θ = 30 degrees.

Therefore, angle Q is 30 degrees.

Since this is the smallest angle in triangle PQS, the answer is 30 degrees.

But wait, let me check the other angles to confirm.

For example, angle at S.

Vectors SP and SQ.

Vector SP = P - S = (1 - 2, √3 - 0) = (-1, √3)

Vector SQ = Q - S = (-2 - 2, 0 - 0) = (-4, 0)

Dot product: (-1)(-4) + (√3)(0) = 4 + 0 = 4

|SP| = sqrt((-1)^2 + (√3)^2) = sqrt(1 + 3) = 2

|SQ| = 4

cosθ = 4 / (2 * 4) = 4 / 8 = 0.5, so θ = 60 degrees.

Angle at P:

Vectors PQ and PS.

Vector PQ = Q - P = (-2 - 1, 0 - √3) = (-3, -√3)

Vector PS = S - P = (2 - 1, 0 - √3) = (1, -√3)

Dot product: (-3)(1) + (-√3)(-√3) = -3 + 3 = 0

Wait, the dot product is zero? That means angle at P is 90 degrees.

So angles are 30°, 60°, 90°. Therefore, the smallest angle is 30°.

Therefore, the answer is 30 degrees.

But wait, the problem says "the degree measure of the smallest angle in triangle PQS". So 30 degrees.

But let me check if this is consistent with the Asymptote code's diagram. Even though the distances between centers don't match the tangency conditions, the coordinates given in the Asymptote code lead to triangle PQS with angles 30°, 60°, 90°.

But since the problem says the circles are tangent to each other and to the sides, maybe the diagram is just illustrative, and the actual answer is 30 degrees.

Alternatively, maybe there's a mistake in assuming the coordinates? But since using the given coordinates leads to a consistent answer, and the problem might have been designed with those coordinates in mind, even if they don't perfectly satisfy the tangency conditions (maybe due to rounding), the answer is 30 degrees.

Therefore, I think the answer is 30 degrees, so the smallest angle is 30°, which is π/6 radians.

**Final Answer**
\boxed{30}

To determine the degree measure of the **smallest angle** in triangle $ PQS $, we begin by analyzing the given coordinates of the centers of the four circles:

- $ P = (1, \sqrt{3}) $
- $ Q = (-2, 0) $
- $ S = (2, 0) $

These coordinates are provided in the Asymptote code, and while they may not perfectly align with the exact geometric conditions of the problem (such as all circles being tangent to each other and the triangle sides), they are used to construct triangle $ PQS $, which is the focus of the question.

---

### Step 1: Compute the Side Lengths of Triangle $ PQS $

We calculate the distances between the points to determine the side lengths of triangle $ PQS $:

- **$ PQ $**:
  $$
  PQ = \sqrt{(1 - (-2))^2 + (\sqrt{3} - 0)^2} = \sqrt{3^2 + (\sqrt{3})^2} = \sqrt{9 + 3} = \sqrt{12} = 2\sqrt{3}
  $$

- **$ QS $**:
  $$
  QS = \sqrt{(2 - (-2))^2 + (0 - 0)^2} = \sqrt{4^2} = 4
  $$

- **$ PS $**:
  $$
  PS = \sqrt{(1 - 2)^2 + (\sqrt{3} - 0)^2} = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
  $$

So, the side lengths of triangle $ PQS $ are:
- $ PS = 2 $
- $ PQ = 2\sqrt{3} $
- $ QS = 4 $

---

### Step 2: Identify the Smallest Angle

In any triangle, the **smallest angle** is opposite the **shortest side**. Here, the shortest side is $ PS = 2 $, so the angle opposite to it, which is **angle $ Q $**, is the smallest angle in triangle $ PQS $.

---

### Step 3: Use the Law of Cosines to Find Angle $ Q $

We apply the Law of Cosines to find the measure of angle $ Q $, which is between sides $ PQ $ and $ QS $, and opposite side $ PS $:

$$
\cos(\angle Q) = \frac{PQ^2 + QS^2 - PS^2}{2 \cdot PQ \cdot QS}
$$

Substitute the known values:

$$
\cos(\angle Q) = \frac{(2\sqrt{3})^2 + 4^2 - 2^2}{2 \cdot 2\sqrt{3} \cdot 4}
= \frac{12 + 16 - 4}{16\sqrt{3}}
= \frac{24}{16\sqrt{3}} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2}
$$

Now, find the angle whose cosine is $ \frac{\sqrt{3}}{2} $:

$$
\angle Q = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = 30^\circ
$$

---

### Final Answer

$$
\boxed{30}
$$

[Assistant]
Explanation:
Okay, so I need to find the degree measure of the smallest angle in triangle PQS. The diagram shows four circles of radius 1 with centers P, Q, R, and S tangent to each other and to the sides of triangle ABC. Let me try to visualize this. There's a triangle ABC with three circles near the corners and one circle somewhere inside? Wait, the Asymptote code shows coordinates for points R, Q, S, and P. Let me check the coordinates given in the Asymptote code:

R is at (0,0), Q is at (-2,0), S is at (2,0), and P is at (1, 1.732). Then points A, B, C are given with coordinates A=(1.366,3.098), B=(-5.73,-1), C=(3.732,-1).

Wait, so maybe triangle ABC is a large triangle with base BC, and points Q, R, S are on the base? Let me think. The centers of the circles are P, Q, R, S. Each circle has radius 1, and they are tangent to each other and to the sides of triangle ABC.

So, since Q, R, S are on the base BC? Let me check their coordinates. R is at (0,0), Q is at (-2,0), S is at (2,0). So they are on the x-axis. Then P is at (1, 1.732). 1.732 is approximately √3, which is about 1.732. So P is above the x-axis.

So maybe triangle ABC has its base BC on the x-axis, with points Q, R, S on BC? But Q is at (-2,0), R at (0,0), S at (2,0). So BC is from B to C, which in the Asymptote code is from (-5.73,-1) to (3.732,-1). Wait, but R is at (0,0), which is above the base BC? Wait, no, BC is at y=-1? Wait, the coordinates of B and C are (-5.73,-1) and (3.732,-1). So BC is a horizontal line at y=-1. Then R is at (0,0), which is above BC. So maybe the triangle ABC is a large triangle with base BC at y=-1, and vertex A somewhere above.

But then the circles are tangent to the sides of ABC. So each circle is tangent to two sides of the triangle? For example, circle Q is tangent to sides AB and BC? Similarly for the others?

But since there are four circles, maybe two circles near the base BC, one near the left side AB, and one near the right side AC? Wait, but the centers are P, Q, R, S. Let me think.

Alternatively, maybe the triangle ABC is such that each side has a circle tangent to it. But there are four circles. Maybe two circles on the base BC, one on AB, and one on AC? But the Asymptote code shows that R is at (0,0), Q at (-2,0), S at (2,0). So maybe Q and S are on the base BC? But BC is at y=-1. Wait, but R is at (0,0), which is above BC. Maybe the circles are tangent to the sides of ABC and to each other.

Alternatively, maybe the triangle ABC is an equilateral triangle? But the coordinates given for A are (1.366, 3.098). Let me check if that's an equilateral triangle. If ABC is equilateral, then all sides are equal. Let me compute the distance between B and C. B is at (-5.73, -1), C is at (3.732, -1). The distance between them is 3.732 - (-5.73) = 9.462 units? Wait, but the coordinates are approximate. Maybe they are using specific values.

Alternatively, maybe the triangle ABC is a 30-60-90 triangle? Or some other triangle. But maybe I don't need to figure out the exact coordinates of ABC. Maybe I can use the positions of the centers P, Q, R, S.

Since all circles have radius 1, and they are tangent to each other. So the distance between centers of tangent circles should be 2 units. Let me check the distances between Q, R, S, and P.

First, Q is at (-2,0), R is at (0,0). The distance between Q and R is 2 units. Since they are both radius 1, so they are tangent. Similarly, R is at (0,0), S is at (2,0), distance is 2 units. So QR and RS are each 2 units. So Q, R, S are colinear on the x-axis, each 2 units apart.

Now, what about the other circles? Let's check the distance between P and Q. P is at (1, √3) approximately. Let me compute the distance between P(1, √3) and Q(-2, 0). The distance squared is (1 - (-2))² + (√3 - 0)² = (3)² + (√3)² = 9 + 3 = 12. So the distance is √12 = 2√3 ≈ 3.464. But since the circles have radius 1, if they are tangent, the distance between centers should be 2. But 2√3 is more than 2, so they are not tangent. Wait, but the problem says the circles are tangent to one another. So maybe my assumption about the coordinates is wrong? Or maybe the diagram is not to scale?

Wait, the Asymptote code is just for drawing, but maybe the coordinates are approximate. Let me check the coordinates again. The Asymptote code defines R=(0,0), Q=(-2,0), S=(2,0), P=(1,1.732). So 1.732 is approximately √3. So P is at (1, √3). Then the distance between P and Q is sqrt[(1 - (-2))² + (√3 - 0)^2] = sqrt[9 + 3] = sqrt[12] = 2*sqrt(3). Which is about 3.464. But if the circles are tangent, the distance between centers should be 2. So that suggests that maybe the diagram is not accurate? Or maybe the circles are tangent to the sides of the triangle, not necessarily to each other? Wait, the problem says "four circles of radius 1 with centers P, Q, R, and S are tangent to one another and to the sides of triangle ABC". So they are tangent to each other and to the sides.

Therefore, the distance between centers of tangent circles should be 2. So if P is tangent to Q, then the distance between P and Q should be 2. But according to coordinates, it's 2√3. So there's a contradiction here. Therefore, maybe the coordinates given in the Asymptote code are not exact? Or maybe I need to figure out the actual coordinates based on the tangency conditions?

Alternatively, maybe the Asymptote code is illustrative, but the actual problem is more general. Maybe I need to solve it without relying on the coordinates?

Alternatively, maybe the triangle ABC is such that the centers P, Q, R, S are located in a specific way. Let me think.

Since there are four circles tangent to each other and to the sides of triangle ABC. This seems similar to the problem of circles tangent to two sides of a triangle and to another circle. Maybe it's related to the Soddy circles or something like that.

Alternatively, since there are four circles, maybe two circles are tangent to the base BC and the other sides, and the other two circles are tangent to the other sides?

Alternatively, maybe triangle ABC is such that there are three circles tangent to two sides and the incircle? But there are four circles here.

Alternatively, maybe the triangle ABC is divided into smaller regions by the circles.

Alternatively, since the problem is about triangle PQS, maybe I need to find coordinates of P, Q, S based on the tangency conditions.

Let me try to think step by step.

First, let me assume that triangle ABC is such that there are four circles of radius 1 tangent to each other and to the sides. Let me try to figure out the positions of P, Q, R, S.

Let me consider that Q, R, S are on the base BC. Since they are tangent to each other and to the sides. If Q and R are on BC, then they are tangent to BC. Similarly, S is also on BC. Wait, but if Q, R, S are all on BC, then they would be tangent to BC, but also tangent to each other. So the distance between Q and R is 2 units (since each has radius 1), same with R and S. So if Q, R, S are on BC, then BC must be long enough to accommodate them.

But if Q is at distance 2 from R, and R is at distance 2 from S, then the total length of BC would be QR + RS = 2 + 2 = 4 units? But then there's also the circle at S, which is tangent to BC and maybe to another side?

Alternatively, maybe there are two circles on the base BC, each tangent to BC and to the other circle, and then another circle tangent to AB and AC? But there are four circles.

Alternatively, maybe there are two circles on each of two sides?

Alternatively, maybe the triangle ABC is such that each side has two circles tangent to it?

This is getting confusing. Maybe I need to think about the configuration.

In some problems, there are circles tangent to two sides of a triangle and to another circle. For example, the mixtilinear incircles. But here, there are four circles.

Alternatively, maybe the triangle ABC is such that there are three circles tangent to two sides and the incircle, and then the incircle itself? But that would be four circles. However, the problem says the circles are tangent to one another and to the sides.

Alternatively, maybe the triangle ABC is divided into smaller triangles by the circles, and the centers form triangle PQS.

Alternatively, since the Asymptote code gives coordinates, maybe I can use those coordinates to calculate the angles? But earlier, the distance between P and Q was 2√3, which is inconsistent with them being tangent. However, maybe the diagram is not to scale?

Wait, the problem says "four circles of radius 1 with centers P, Q, R, and S are tangent to one another and to the sides of triangle ABC". So if they are tangent to each other, then the distance between centers should be 2. So if in the Asymptote code, the distance between P and Q is 2√3, which is more than 2, that suggests that maybe the diagram is not accurate? Or maybe the coordinates are approximate?

Alternatively, maybe the Asymptote code is using specific coordinates for illustration, but the actual problem requires a different approach.

Alternatively, maybe I can use coordinates to find the positions of P, Q, R, S based on the tangency conditions. Let me try that.

Let me assume that triangle ABC is such that there are four circles of radius 1 tangent to each other and to the sides. Let me try to model this.

Let me consider that Q, R, S are on the base BC. Let me assume that BC is horizontal. Let me suppose that Q is tangent to BC and to AB, R is tangent to BC and to AC, and S is tangent to BC and to... Wait, but there are four circles. Maybe there is another circle tangent to AB and AC?

Alternatively, maybe there are two circles on the base BC, each tangent to BC and to the other circle, and then two circles tangent to AB and AC?

Alternatively, maybe the configuration is similar to the problem of circles tangent to two sides and to another circle.

Alternatively, since there are four circles, maybe two circles are tangent to the base BC and to the other two sides, and the other two circles are tangent to the other sides?

This is getting too vague. Maybe I need to think about the triangle ABC.

Alternatively, since the Asymptote code gives coordinates for A, B, C, maybe I can use those coordinates to find the positions of P, Q, R, S?

Wait, the Asymptote code defines:

R=(0,0);

Q=(-2,0);

S=(2,0);

P=(1,1.732);

B=(-5.73,-1);

C=(3.732,-1);

A=(1.366,3.098);

So, if I can use these coordinates, even though the distances between centers may not be exactly 2, maybe I can compute the angles of triangle PQS?

But the problem says the circles are tangent to each other and to the sides. So maybe the coordinates are approximate?

Alternatively, maybe the coordinates are exact? Let me check.

First, let me check the coordinates of P=(1, 1.732). 1.732 is approximately √3, which is about 1.732. So maybe P is at (1, √3). Similarly, Q is at (-2, 0), R at (0,0), S at (2,0).

So if I take these coordinates as exact, then let me compute the distances between P, Q, R, S.

Distance between Q(-2, 0) and R(0,0) is 2 units. Good, since they are tangent.

Distance between R(0,0) and S(2,0) is 2 units. Good.

Distance between Q(-2,0) and S(2,0) is 4 units.

Distance between P(1, √3) and Q(-2,0): sqrt[(1 - (-2))² + (√3 - 0)^2] = sqrt[9 + 3] = sqrt[12] = 2*sqrt(3) ≈ 3.464.

Distance between P(1, √3) and R(0,0): sqrt[(1 - 0)^2 + (√3 - 0)^2] = sqrt[1 + 3] = sqrt[4] = 2. So that's good! So P is tangent to R.

Similarly, distance between P(1, √3) and S(2,0): sqrt[(1 - 2)^2 + (√3 - 0)^2] = sqrt[1 + 3] = 2. So P is tangent to S as well.

So P is tangent to Q? Wait, no. Wait, P is tangent to R and S, but not to Q? Wait, the distance between P and Q is 2√3, which is more than 2, so they are not tangent. But the problem says all four circles are tangent to one another. Wait, so maybe there's a mistake here?

Wait, the problem says "four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of triangle ABC". So all four circles are tangent to each other. So each pair of circles should be tangent. But according to the coordinates, P is tangent to R and S, but not to Q. Similarly, Q is tangent to R, R is tangent to S, but Q is not tangent to S? Wait, distance between Q(-2,0) and S(2,0) is 4 units, which is more than 2, so they are not tangent.

So there's a contradiction here. Therefore, the Asymptote code's coordinates must be illustrative, not exact. Therefore, I can't rely on them for exact distances.

Therefore, I need to figure out the positions of P, Q, R, S based on the tangency conditions.

Let me try again. Let me assume that triangle ABC is such that there are four circles of radius 1 tangent to each other and to the sides. Let me think about the configuration.

In such problems, often the centers of the circles lie along the angle bisectors of the triangle. However, with four circles, it's more complex.

Alternatively, maybe the triangle ABC is such that there are two circles tangent to the base BC and to the other sides, and two circles tangent to the other sides?

Alternatively, since there are four circles, maybe two circles are tangent to the base BC and to the other two sides, and the other two circles are tangent to the other sides?

Alternatively, maybe the triangle ABC is divided into smaller triangles by the circles, and the centers form a smaller triangle.

Alternatively, since the problem is about triangle PQS, maybe I can find coordinates for P, Q, S based on the tangency conditions.

Let me consider that Q, R, S are on the base BC. Let me assume that BC is horizontal. Let me suppose that Q and S are on BC, each tangent to BC and to the other circles. Also, R is between Q and S? Wait, but if there are three circles on BC, each tangent to BC and to each other, then the distance between Q and R is 2, and between R and S is 2, so total length of BC would be QR + RS = 4. But then there's also the circle P, which is tangent to Q, R, and S? Wait, but if P is tangent to Q, R, and S, then it's the circumcircle? But all circles have radius 1.

Alternatively, maybe P is tangent to Q and S, and also tangent to another side?

Alternatively, maybe the triangle ABC is such that there are two circles on the base BC, each tangent to BC and to the other circle, and then two circles tangent to the other sides.

Alternatively, maybe the triangle ABC is such that there are three circles tangent to two sides and the incircle, and the incircle itself? But that would be four circles.

Alternatively, maybe the triangle ABC is equilateral? If ABC is equilateral, then maybe the circles are arranged symmetrically.

Let me try assuming ABC is equilateral. Let me suppose ABC is equilateral with side length 'a'. Then, the inradius is (a√3)/6. If there are circles of radius 1 tangent to the sides, then maybe the inradius is related. However, with four circles, it's more complex.

Alternatively, maybe the triangle ABC is a right triangle?

Alternatively, since the Asymptote code has coordinates for A, B, C, maybe I can use those coordinates to find the positions of P, Q, R, S?

Wait, let me check the coordinates again.

In the Asymptote code:

A=(1.366,3.098)

B=(-5.73,-1)

C=(3.732,-1)

So, coordinates of B and C are at y=-1, so BC is horizontal. The coordinates of A are approximately (1.366, 3.098). Let me check if ABC is a specific triangle.

Let me compute the lengths of AB, BC, and AC.

First, BC: distance between B(-5.73, -1) and C(3.732, -1) is 3.732 - (-5.73) = 9.462 units.

AB: distance between A(1.366, 3.098) and B(-5.73, -1). Let me compute:

Δx = 1.366 - (-5.73) = 7.096

Δy = 3.098 - (-1) = 4.098

Distance AB = sqrt(7.096² + 4.098²). Let me approximate:

7.096² ≈ 50.35

4.098² ≈ 16.79

Total ≈ 67.14, so sqrt(67.14) ≈ 8.194

Similarly, AC: distance between A(1.366, 3.098) and C(3.732, -1):

Δx = 3.732 - 1.366 = 2.366

Δy = -1 - 3.098 = -4.098

Distance AC = sqrt(2.366² + 4.098²) ≈ sqrt(5.598 + 16.79) ≈ sqrt(22.388) ≈ 4.732

So sides are approximately BC ≈ 9.46, AB ≈ 8.19, AC ≈ 4.73. So triangle ABC is scalene.

Now, the centers of the circles are P, Q, R, S. Let me check their coordinates:

R=(0,0), Q=(-2,0), S=(2,0), P=(1,1.732).

So, if I consider the triangle ABC with coordinates as given, then the circles are located as follows:

- Q is at (-2, 0), which is inside the triangle? Let me check if that's possible.

Point B is at (-5.73, -1), so moving from B to C, which is at (3.732, -1). The base BC is from x=-5.73 to x=3.732 at y=-1. The point Q is at (-2, 0), which is above the base BC. Similarly, R is at (0,0), also above BC, and S is at (2,0), also above BC. P is at (1, 1.732), which is above R and S.

So maybe these circles are located near the sides of the triangle. For example, circle Q is tangent to AB and BC, circle R is tangent to BC and... maybe another side?

Alternatively, since the circles are tangent to the sides of ABC, each circle is tangent to two sides. For example, circle Q is tangent to AB and BC, circle R is tangent to BC and... maybe AC? Wait, but there are four circles.

Alternatively, maybe two circles are tangent to BC and the other sides, and two circles are tangent to AB and AC?

Alternatively, since there are four circles, maybe each circle is tangent to two sides and to another circle.

Alternatively, since the problem says they are tangent to one another and to the sides, so each circle is tangent to two sides of the triangle and to two other circles.

But with four circles, it's complex.

Alternatively, maybe the triangle ABC is such that there are two circles tangent to AB and BC, two circles tangent to BC and AC, and one circle tangent to AB and AC? But that would be five circles.

Alternatively, maybe the four circles are arranged such that two are on the base BC, each tangent to BC and to the other circle, and the other two are tangent to AB and AC.

But how would they be tangent to each other?

Alternatively, maybe the centers P, Q, R, S form a specific quadrilateral.

But since the problem is about triangle PQS, maybe I can find coordinates of P, Q, S based on the tangency conditions.

Let me try to think about circle Q. If it's tangent to BC and AB, then its center Q is located at a distance of 1 from BC and from AB. Similarly for other circles.

Given that BC is the base at y=-1, then the distance from Q to BC is 1. Since BC is at y=-1, the y-coordinate of Q should be -1 + 1 = 0. Which matches the coordinates of Q at (-2, 0). Similarly, R is at (0,0), so distance from BC is 1. Similarly for S at (2,0).

So Q, R, S are all at y=0, which is 1 unit above BC.

Now, circle Q is also tangent to AB. So the distance from Q to AB is 1. Similarly for circle R and AC?

Let me find the equation of AB to check.

Point A is at (1.366, 3.098), point B is at (-5.73, -1). Let me compute the equation of line AB.

First, compute the slope of AB:

m = (y_A - y_B)/(x_A - x_B) = (3.098 - (-1))/(1.366 - (-5.73)) = (4.098)/(7.096) ≈ 0.577.

Which is approximately 1/√3 ≈ 0.577. So the slope is approximately 1/√3.

Therefore, the equation of AB can be written as y - y_B = m(x - x_B).

So y - (-1) = (1/√3)(x - (-5.73)) => y + 1 = (1/√3)(x + 5.73).

But since the distance from Q(-2, 0) to AB is 1, let me verify that.

The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²).

First, let me write the equation of AB in standard form.

Starting from y + 1 = (1/√3)(x + 5.73). Multiply both sides by √3:

√3(y + 1) = x + 5.73

Rearranged: x - √3 y + (5.73 - √3) = 0.

Wait, let me check:

Starting from y + 1 = (1/√3)(x + 5.73)

Multiply both sides by √3:

√3 y + √3 = x + 5.73

Bring all terms to left:

x - √3 y + (5.73 - √3) = 0

So the standard form is x - √3 y + (5.73 - √3) = 0.

Now, the distance from Q(-2, 0) to this line is |(-2) - √3*(0) + (5.73 - √3)| / sqrt(1 + (√3)^2) = | -2 + 5.73 - √3 | / sqrt(1 + 3) = |3.73 - √3| / 2.

Compute 3.73 - √3 ≈ 3.73 - 1.732 ≈ 1.998 ≈ 2. So the distance is approximately 2 / 2 = 1. Which matches the required distance of 1.

Therefore, the distance from Q to AB is indeed 1, which means circle Q is tangent to AB and BC. Similarly, circle S is tangent to AC and BC.

Now, what about circle R? It is at (0,0), which is 1 unit above BC. Let me check if it is tangent to AC.

First, find the equation of AC.

Point A is (1.366, 3.098), point C is (3.732, -1).

Slope of AC: m = (-1 - 3.098)/(3.732 - 1.366) = (-4.098)/2.366 ≈ -1.732. Which is approximately -√3.

So the equation of AC is y - y_C = m(x - x_C).

So y - (-1) = -√3(x - 3.732) => y + 1 = -√3 x + 3.732√3.

Rearranged: √3 x + y + (1 - 3.732√3) = 0.

Now, compute the distance from R(0,0) to this line.

Distance = |√3*0 + 1*0 + (1 - 3.732√3)| / sqrt( (√3)^2 + 1^2 ) = |1 - 3.732√3| / 2.

Compute 3.732√3 ≈ 3.732 * 1.732 ≈ 6.464. So 1 - 6.464 ≈ -5.464. Absolute value is 5.464. Divided by 2 is ≈ 2.732. Which is not equal to 1.

Wait, that's a problem. The distance from R to AC is approximately 2.732, but it should be 1 since the circle is tangent to AC. Therefore, there's inconsistency here.

But according to the Asymptote code, R is at (0,0), which is supposed to be the center of a circle tangent to the sides. So maybe my calculation is wrong?

Wait, let me check the equation of AC again.

Point A is (1.366, 3.098), point C is (3.732, -1).

Let me compute the slope again:

m = (y_C - y_A)/(x_C - x_A) = (-1 - 3.098)/(3.732 - 1.366) = (-4.098)/2.366 ≈ -1.732. Which is -√3.

So the slope is -√3.

Equation of AC: y - y_A = m(x - x_A).

So y - 3.098 = -√3(x - 1.366).

Let me rearrange this:

y = -√3 x + √3 * 1.366 + 3.098.

Compute √3 * 1.366 ≈ 1.732 * 1.366 ≈ 2.366.

So y ≈ -√3 x + 2.366 + 3.098 ≈ -√3 x + 5.464.

Therefore, the equation of AC is y + √3 x - 5.464 = 0.

So standard form: √3 x + y - 5.464 = 0.

Now, distance from R(0,0) to this line is |√3*0 + 1*0 - 5.464| / sqrt( (√3)^2 + 1^2 ) = | -5.464 | / 2 ≈ 5.464 / 2 ≈ 2.732.

Which is not 1. So this suggests that circle R is not tangent to AC. But according to the problem statement, all circles are tangent to the sides. Therefore, there must be an error in my assumption.

Alternatively, maybe circle R is tangent to AB and BC? But then it would be similar to Q. But Q is already tangent to AB and BC.

Alternatively, maybe circle R is tangent to BC and another side?

Wait, if R is at (0,0), which is 1 unit above BC, and if it's tangent to another side, say AB or AC. But we saw that distance to AB is... Let me check distance from R(0,0) to AB.

Earlier, the equation of AB was x - √3 y + (5.73 - √3) = 0.

Wait, earlier I had:

From AB: y + 1 = (1/√3)(x + 5.73).

Which rearranged to x - √3 y + (5.73 - √3) = 0.

So distance from R(0,0) to AB is |0 - √3*0 + 5.73 - √3| / 2 = |5.73 - √3| / 2 ≈ |5.73 - 1.732| / 2 ≈ 3.998 / 2 ≈ 1.999 ≈ 2.

Which is not 1. So circle R is not tangent to AB.

So this suggests that the Asymptote code's coordinates are not consistent with the problem's conditions. Therefore, I need to find another approach.

Let me try to think of the configuration again.

Assume that there are four circles of radius 1, tangent to each other and to the sides of triangle ABC. Let me consider that two of the circles are on the base BC, each tangent to BC and to each other, and the other two circles are tangent to the other sides.

Let me denote the two circles on BC as Q and R, with centers at distance 2 units apart. Let me assume that the base BC is horizontal, and the circles Q and R are located above BC, each tangent to BC. Then, the centers Q and R are at height 1 above BC.

Now, there are two more circles, P and S, which are tangent to the other sides. Let me assume that circle P is tangent to AB and AC, and circle S is tangent to AB and AC? No, that would be two circles.

Alternatively, maybe circle P is tangent to AB and BC, and circle S is tangent to AC and BC? But then we already have Q and R on BC.

Wait, if there are four circles, maybe two on BC (Q and R), one on AB (P), and one on AC (S)?

But then how are they tangent to each other?

Alternatively, maybe circle P is tangent to AB, AC, and Q, and circle S is tangent to AB, AC, and R?

But this is getting too vague.

Alternatively, since the problem mentions triangle PQS, maybe I can find the coordinates of P, Q, S based on the tangency conditions.

Let me consider that Q and R are on BC, each at distance 1 from BC, and distance 2 apart. Let me assume BC is horizontal, and let me place coordinate system such that BC is on the x-axis, with B at (0,0), C at (c, 0), and Q at (q, 1), R at (r, 1). Since they are tangent to each other, the distance between Q and R is 2, so |r - q| = 2.

But also, they are tangent to BC, so their y-coordinate is 1.

Now, there are other circles. Let me assume that there is a circle P tangent to AB, AC, and Q. Similarly, a circle S tangent to AB, AC, and R?

Alternatively, maybe there is a circle tangent to AB, BC, and Q, and another circle tangent to AC, BC, and R?

But then there would be four circles: Q, R, P, S.

Let me try this.

Let me assume that circle Q is tangent to AB, BC, and circle R. Similarly, circle R is tangent to AC, BC, and circle Q. Wait, but then there would be two circles on BC, each tangent to one side and the base.

Alternatively, maybe circle Q is tangent to AB, BC, and circle P, and circle R is tangent to AC, BC, and circle P?

This is getting too complex. Maybe I need to use coordinate geometry.

Let me try to set up coordinates. Let me assume that BC is on the x-axis, with B at (0,0), C at (c, 0). Let me assume that circle Q is tangent to BC and AB, and circle R is tangent to BC and AC. Also, circles Q and R are tangent to each other.

Let me denote the coordinates of Q as (x1, 1), since it's 1 unit above BC. Similarly, R is at (x2, 1). The distance between Q and R is 2, so sqrt[(x2 - x1)^2 + (1 - 1)^2] = |x2 - x1| = 2. So x2 = x1 + 2 or x1 = x2 + 2.

Now, circle Q is tangent to AB. Let me find the equation of AB. Let me assume that A is at some point (a, b).

But since circle Q is tangent to AB and BC, and has center at (x1, 1), the distance from Q to AB is 1. Similarly, the distance from Q to BC is 1, which is already satisfied since y-coordinate is 1.

Similarly for circle R.

Let me find the coordinates of A. Since circle Q is tangent to AB, and circle R is tangent to AC, and the triangle ABC has AB and AC as sides.

This seems too abstract. Maybe I need to relate the coordinates of A with the positions of Q and R.

Alternatively, since there are four circles, maybe the triangle ABC is such that it has an incircle and three excircles? No, that's not helpful.

Alternatively, since there are four circles, maybe it's a special triangle where the circles are arranged symmetrically.

Wait, going back to the Asymptote code, even though the distances between P and Q are not 2, maybe I can use the coordinates to calculate the angles of triangle PQS?

Given that in the Asymptote code, P is at (1, √3), Q is at (-2, 0), S is at (2, 0). Let me compute the lengths of the sides of triangle PQS.

First, PQ: distance between P(1, √3) and Q(-2, 0):

sqrt[(1 - (-2))² + (√3 - 0)²] = sqrt[9 + 3] = sqrt[12] = 2*sqrt(3) ≈ 3.464

QS: distance between Q(-2, 0) and S(2, 0): sqrt[(2 - (-2))² + (0 - 0)^2] = sqrt[16] = 4

PS: distance between P(1, √3) and S(2, 0):

sqrt[(1 - 2)^2 + (√3 - 0)^2] = sqrt[1 + 3] = sqrt[4] = 2

So triangle PQS has sides of length 2, 2√3, and 4.

Wait, but in a triangle, the sum of any two sides must be greater than the third. Let's check:

2 + 2√3 ≈ 2 + 3.464 = 5.464 > 4

2 + 4 = 6 > 2√3 ≈ 3.464

2√3 + 4 ≈ 3.464 + 4 = 7.464 > 2

So it's a valid triangle.

Now, to find the smallest angle in triangle PQS. The smallest angle is opposite the shortest side. The sides are 2, 2√3, and 4. The shortest side is 2, which is PS. Therefore, the smallest angle is the angle opposite PS, which is angle Q.

Wait, no. Wait, in triangle PQS, the sides are:

PQ = 2√3 ≈ 3.464

QS = 4

PS = 2

So the sides are PS = 2, PQ = 2√3, QS = 4.

Therefore, the smallest side is PS = 2, so the angle opposite to it is angle Q. Wait, no. In triangle PQS, the side opposite angle P is QS = 4, opposite angle Q is PS = 2, and opposite angle S is PQ = 2√3.

Therefore, the smallest angle is angle Q, which is opposite the shortest side PS = 2.

Wait, but let me confirm. In triangle PQS, the angles are at P, Q, S.

Side opposite angle P is QS = 4

Side opposite angle Q is PS = 2

Side opposite angle S is PQ = 2√3

Therefore, the smallest angle is angle Q.

But the question asks for the degree measure of the smallest angle in triangle PQS. So I need to find angle Q.

Alternatively, maybe I can use the Law of Cosines to find the angles.

Let me compute angle at Q.

In triangle PQS, angle at Q is between sides QP and QS.

Wait, sides adjacent to angle Q are QP and QS. Wait, no. Wait, angle at Q is between sides QP and QR? Wait, no, triangle PQS has vertices P, Q, S. So angle at Q is between sides QP and QS.

Wait, the sides adjacent to angle Q are QP and QS. Wait, but QS is the side from Q to S, which is length 4, and QP is from Q to P, which is length 2√3. The side opposite angle Q is PS, which is length 2.

So using the Law of Cosines on angle Q:

PS² = QP² + QS² - 2 * QP * QS * cos(angle Q)

So:

2² = (2√3)² + 4² - 2 * (2√3) * 4 * cos(angle Q)

Compute:

4 = 12 + 16 - 16√3 * cos(angle Q)

Simplify:

4 = 28 - 16√3 * cos(angle Q)

Rearranged:

16√3 * cos(angle Q) = 28 - 4 = 24

Therefore:

cos(angle Q) = 24 / (16√3) = (3) / (2√3) = √3 / 2

Therefore, angle Q is arccos(√3 / 2) = 30 degrees.

So the smallest angle in triangle PQS is 30 degrees.

But wait, let me check if this is correct.

Alternatively, since the coordinates are given, maybe I can compute the angle using coordinates.

Coordinates:

P(1, √3), Q(-2, 0), S(2, 0)

Vector QP = P - Q = (1 - (-2), √3 - 0) = (3, √3)

Vector QS = S - Q = (2 - (-2), 0 - 0) = (4, 0)

The angle at Q is between vectors QP and QS.

The formula for the angle between two vectors u and v is:

cosθ = (u • v) / (|u| |v|)

Compute the dot product of QP and QS:

(3)(4) + (√3)(0) = 12 + 0 = 12

|QP| = sqrt(3² + (√3)^2) = sqrt(9 + 3) = sqrt(12) = 2√3

|QS| = sqrt(4² + 0²) = 4

Therefore, cosθ = 12 / (2√3 * 4) = 12 / (8√3) = 3 / (2√3) = √3 / 2

Which gives θ = 30 degrees.

Therefore, angle Q is 30 degrees.

Since this is the smallest angle in triangle PQS, the answer is 30 degrees.

But wait, let me check the other angles to confirm.

For example, angle at S.

Vectors SP and SQ.

Vector SP = P - S = (1 - 2, √3 - 0) = (-1, √3)

Vector SQ = Q - S = (-2 - 2, 0 - 0) = (-4, 0)

Dot product: (-1)(-4) + (√3)(0) = 4 + 0 = 4

|SP| = sqrt((-1)^2 + (√3)^2) = sqrt(1 + 3) = 2

|SQ| = 4

cosθ = 4 / (2 * 4) = 4 / 8 = 0.5, so θ = 60 degrees.

Angle at P:

Vectors PQ and PS.

Vector PQ = Q - P = (-2 - 1, 0 - √3) = (-3, -√3)

Vector PS = S - P = (2 - 1, 0 - √3) = (1, -√3)

Dot product: (-3)(1) + (-√3)(-√3) = -3 + 3 = 0

Wait, the dot product is zero? That means angle at P is 90 degrees.

So angles are 30°, 60°, 90°. Therefore, the smallest angle is 30°.

Therefore, the answer is 30 degrees.

But wait, the problem says "the degree measure of the smallest angle in triangle PQS". So 30 degrees.

But let me check if this is consistent with the Asymptote code's diagram. Even though the distances between centers don't match the tangency conditions, the coordinates given in the Asymptote code lead to triangle PQS with angles 30°, 60°, 90°.

But since the problem says the circles are tangent to each other and to the sides, maybe the diagram is just illustrative, and the actual answer is 30 degrees.

Alternatively, maybe there's a mistake in assuming the coordinates? But since using the given coordinates leads to a consistent answer, and the problem might have been designed with those coordinates in mind, even if they don't perfectly satisfy the tangency conditions (maybe due to rounding), the answer is 30 degrees.

Therefore, I think the answer is 30 degrees, so the smallest angle is 30°, which is π/6 radians.

**Final Answer**
\boxed{30}

To determine the degree measure of the **smallest angle** in triangle $ PQS $, we begin by analyzing the given coordinates of the centers of the four circles:

- $ P = (1, \sqrt{3}) $
- $ Q = (-2, 0) $
- $ S = (2, 0) $

These coordinates are provided in the Asymptote code, and while they may not perfectly align with the exact geometric conditions of the problem (such as all circles being tangent to each other and the triangle sides), they are used to construct triangle $ PQS $, which is the focus of the question.

---

### Step 1: Compute the Side Lengths of Triangle $ PQS $

We calculate the distances between the points to determine the side lengths of triangle $ PQS $:

- **$ PQ $**:
  $$
  PQ = \sqrt{(1 - (-2))^2 + (\sqrt{3} - 0)^2} = \sqrt{3^2 + (\sqrt{3})^2} = \sqrt{9 + 3} = \sqrt{12} = 2\sqrt{3}
  $$

- **$ QS $**:
  $$
  QS = \sqrt{(2 - (-2))^2 + (0 - 0)^2} = \sqrt{4^2} = 4
  $$

- **$ PS $**:
  $$
  PS = \sqrt{(1 - 2)^2 + (\sqrt{3} - 0)^2} = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
  $$

So, the side lengths of triangle $ PQS $ are:
- $ PS = 2 $
- $ PQ = 2\sqrt{3} $
- $ QS = 4 $

---

### Step 2: Identify the Smallest Angle

In any triangle, the **smallest angle** is opposite the **shortest side**. Here, the shortest side is $ PS = 2 $, so the angle opposite to it, which is **angle $ Q $**, is the smallest angle in triangle $ PQS $.

---

### Step 3: Use the Law of Cosines to Find Angle $ Q $

We apply the Law of Cosines to find the measure of angle $ Q $, which is between sides $ PQ $ and $ QS $, and opposite side $ PS $:

$$
\cos(\angle Q) = \frac{PQ^2 + QS^2 - PS^2}{2 \cdot PQ \cdot QS}
$$

Substitute the known values:

$$
\cos(\angle Q) = \frac{(2\sqrt{3})^2 + 4^2 - 2^2}{2 \cdot 2\sqrt{3} \cdot 4}
= \frac{12 + 16 - 4}{16\sqrt{3}}
= \frac{24}{16\sqrt{3}} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2}
$$

Now, find the angle whose cosine is $ \frac{\sqrt{3}}{2} $:

$$
\angle Q = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = 30^\circ
$$

---

### Final Answer

$$
\boxed{30}
$$

Solution:
Join $PQ$, $PR$, $PS$, $RQ$, and $RS$. Since the circles with center $Q$, $R$ and $S$ are all tangent to $BC$, then $QR$ and $RS$ are each parallel to $BC$ (as the centres $Q$, $R$ and $S$ are each 1 unit above $BC$). This tells us that $QS$ passes through $R$. When the centers of tangent circles are joined, the line segments formed pass through the associated point of tangency, and so have lengths equal to the sum of the radii of those circles. Therefore, $QR=RS=PR=PS=1+1=2$.

[asy]
size(200);
pair P, Q, R, S;
Q=(0,0);
R=(2,0);
S=(4,0);
P=(3,1.732);
label("Q", Q, SW);
label("R", R, dir(270));
label("S", S, SE);
label("P", P, N);
draw(circle(Q,1), dashed);
draw(circle(P,1), dashed);
draw(circle(R,1), dashed);
draw(circle(S,1), dashed);
draw(P--Q--S--P--R);
[/asy]

Since $PR=PS=RS$, we know $\triangle PRS$ is equilateral, so $\angle PSR=\angle PRS=60^\circ$.  Since $\angle PRS=60^\circ$ and $QRS$ is a straight line, we have $\angle QRP=180^\circ-60^\circ=120^\circ$. Since $QR=RP$, we know $\triangle QRP$ is isosceles, so $$\angle PQR = \frac{1}{2}(180^\circ-120^\circ)= 30^\circ.$$Since $\angle PQS=30^\circ$ and $\angle PSQ=60^\circ$, we have $\angle QPS = 180^\circ - 30^\circ - 60^\circ = 90^\circ$, so $\triangle PQS$ is a $30^\circ$-$60^\circ$-$90^\circ$ triangle. Thus, the answer is $\boxed{30^\circ}$.


Explanation:
Okay, so I need to figure out how many positive two-digit integers are factors of both 100 and 150. Hmm, let me start by recalling what it means for a number to be a factor of another number. A factor of a number is an integer that divides that number without leaving a remainder. So, if a number is a factor of both 100 and 150, it must divide both of them exactly.

First, maybe I should find the common factors of 100 and 150. I remember that the common factors of two numbers are the factors of their greatest common divisor (GCD). So, if I can find the GCD of 100 and 150, then all the factors of that GCD will be the common factors of 100 and 150. That seems like a good approach. Let me try that.

To find the GCD of 100 and 150, I can use the Euclidean algorithm. Let me recall how that works. The Euclidean algorithm involves dividing the larger number by the smaller one, then replacing the larger number with the smaller number and the smaller number with the remainder from the division, and repeating this process until the remainder is zero. The last non-zero remainder is the GCD.

So, let's apply that to 100 and 150.

First, divide 150 by 100. The quotient is 1 and the remainder is 50. So, GCD(150, 100) = GCD(100, 50).

Now, divide 100 by 50. The quotient is 2 and the remainder is 0. Since the remainder is 0, the GCD is the last non-zero remainder, which is 50.

Therefore, the GCD of 100 and 150 is 50. That means all common factors of 100 and 150 are the factors of 50.

So, now I need to find all the positive factors of 50 and then determine which of those are two-digit numbers.

Let me list the factors of 50. To do that, I can start by prime factorizing 50.

50 can be broken down into prime factors as follows: 50 = 2 × 25 = 2 × 5².

The exponents of the prime factors are 1 for 2 and 2 for 5. To find the number of factors, we add 1 to each exponent and multiply them. So, (1+1)(2+1) = 2×3 = 6 factors.

But I don't need the number of factors, I need the actual factors. Let me list them.

Starting with 1, then 2, 5, 10, 25, and 50. Wait, let me check that.

Factors of 50 are numbers that divide 50 without a remainder. Let me list them systematically:

1 × 50 = 50
2 × 25 = 50
5 × 10 = 50

So, the factors are 1, 2, 5, 10, 25, 50.

So, those are all the positive factors of 50. Now, the question asks for two-digit integers. Let me check which of these factors are two-digit numbers.

Looking at the list: 1 is a single-digit number. 2 is single-digit. 5 is single-digit. 10 is two-digit. 25 is two-digit. 50 is two-digit.

So, the two-digit factors are 10, 25, and 50.

Wait, but hold on. Let me confirm that these numbers are indeed factors of both 100 and 150.

Starting with 10: 100 divided by 10 is 10, which is an integer. 150 divided by 10 is 15, which is also an integer. So, 10 is a common factor.

Next, 25: 100 divided by 25 is 4, which is an integer. 150 divided by 25 is 6, which is also an integer. So, 25 is a common factor.

Then, 50: 100 divided by 50 is 2, which is an integer. 150 divided by 50 is 3, which is also an integer. So, 50 is a common factor.

Therefore, there are three two-digit numbers that are factors of both 100 and 150.

But wait, let me make sure I didn't miss any factors. Let me check if there are any other factors of 50 that I might have missed.

Earlier, I listed 1, 2, 5, 10, 25, 50. Let me check if there are more.

Well, 50 divided by 1 is 50, which we have. Divided by 2 is 25, which we have. Divided by 5 is 10, which we have. Divided by 10 is 5, which is already listed. Divided by 25 is 2, already listed. Divided by 50 is 1, already listed. So, no other factors.

Therefore, only 10, 25, and 50 are the two-digit factors.

But hold on, let me think again. Is there a possibility that there are other common factors that aren't factors of 50? Wait, no. Because the GCD is 50, so all common factors must divide 50. Therefore, there can't be any other common factors outside of the factors of 50.

Therefore, my conclusion is that there are three two-digit integers that are factors of both 100 and 150.

But just to be thorough, let me check if there are any other numbers between 10 and 99 that divide both 100 and 150.

For example, let's take 20. Does 20 divide both 100 and 150? 100 divided by 20 is 5, which is okay. 150 divided by 20 is 7.5, which is not an integer. So, 20 is not a common factor.

How about 15? 100 divided by 15 is approximately 6.666..., which is not an integer. So, 15 is not a factor of 100.

What about 25? We already checked that.

How about 50? Also checked.

What about 10? Checked.

What about 2? No, single-digit.

What about 5? Single-digit.

What about 1? Single-digit.

How about 3? 100 divided by 3 is not an integer. 150 divided by 3 is 50, but since 3 isn't a factor of 100, it's not a common factor.

Similarly, 4? 100 divided by 4 is 25, but 150 divided by 4 is 37.5, not integer.

So, 4 isn't a common factor.

How about 6? 100 divided by 6 is not integer.

7? 100 divided by 7 is not integer.

8? 100 divided by 8 is 12.5, not integer.

9? 100 divided by 9 is not integer.

So, none of these.

How about 10, 25, 50.

Wait, what about 50? 50 is a two-digit number.

Is there another number? Let me check 25.

What about numbers like 10, 25, 50.

Is there another? Let me check 10, 25, 50.

Wait, 50 is two-digit. 25 is two-digit. 10 is two-digit.

Is there another? Let me check 50.

Wait, 50 is the largest factor.

Wait, maybe I can check the factors of 100 and 150 individually and see their common factors.

Let me try that approach as a cross-check.

First, list all factors of 100.

Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.

Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.

Now, common factors of 100 and 150 are the numbers that appear in both lists.

Looking at the factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.

Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.

Common factors are numbers that are in both. Let me compare them:

1 is common.
2 is common.
4 is not in the factors of 150.
5 is common.
10 is common.
20 is not in factors of 150.
25 is common.
50 is common.
100 is not in factors of 150.

So, the common factors are 1, 2, 5, 10, 25, 50.

Which matches the factors of 50, as we found earlier.

Therefore, the common factors are indeed 1, 2, 5, 10, 25, 50.

So, among these, the two-digit numbers are 10, 25, 50.

Therefore, there are three such numbers.

So, my answer is 3.

But let me check once more to make sure I didn't make a mistake.

Is there a possibility that I missed a factor? For example, is there a two-digit number that divides both 100 and 150 but isn't in the list?

Looking at the common factors, they are 1, 2, 5, 10, 25, 50.

So, two-digit numbers are 10, 25, 50.

Yes, that's three numbers.

Alternatively, maybe I can think of the problem as finding the intersection of the two-digit factors of 100 and the two-digit factors of 150.

Let me try that approach.

First, find all two-digit factors of 100.

Factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100.

Two-digit factors are 10, 20, 25, 50.

Similarly, factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.

Two-digit factors are 10, 15, 25, 30, 50.

Now, the intersection of these two sets (two-digit factors of 100 and two-digit factors of 150) is {10, 25, 50}.

Therefore, again, three numbers.

So, this confirms the same result.

Therefore, the answer is 3.

But just to be absolutely thorough, let me check each of these numbers again.

10: divides 100? Yes, 100/10=10. Divides 150? 150/10=15. Yes.

25: 100/25=4, 150/25=6. Yes.

50: 100/50=2, 150/50=3. Yes.

So, all three are valid.

Is there any other number? Let me check if there are other two-digit numbers that divide both.

For example, 15: divides 150? Yes, 150/15=10. Divides 100? 100/15≈6.666... No. So, not a factor.

20: divides 100? Yes, 100/20=5. Divides 150? 150/20=7.5. No.

30: divides 150? Yes, 150/30=5. Divides 100? 100/30≈3.333... No.

So, no.

How about 5? Single-digit.

So, no.

Therefore, there are exactly three two-digit numbers.

Therefore, the answer is 3.

**Final Answer**
The number of positive two-digit integers that are factors of both 100 and 150 is \boxed{3}.

To determine how many **positive two-digit integers** are factors of **both 100 and 150**, we begin by identifying the **common factors** of these two numbers.

---

### Step 1: Find the Greatest Common Divisor (GCD)

We use the **Euclidean algorithm** to find the GCD of 100 and 150:

- $ 150 \div 100 = 1 $ remainder $ 50 $
- $ 100 \div 50 = 2 $ remainder $ 0 $

So, the GCD of 100 and 150 is **50**.

---

### Step 2: Find All Factors of the GCD

Now, we find all **positive factors** of 50:

- $ 50 = 2 \times 5^2 $
- The factors are: $ 1, 2, 5, 10, 25, 50 $

---

### Step 3: Identify Two-Digit Factors

From the list of factors of 50, we select the **two-digit numbers**:

- $ 10, 25, 50 $

These are the only two-digit numbers that divide both 100 and 150.

---

### Step 4: Verify Each Candidate

Let’s confirm that each of these numbers is indeed a factor of both 100 and 150:

- $ 100 \div 10 = 10 $, $ 150 \div 10 = 15 $ → ✅
- $ 100 \div 25 = 4 $, $ 150 \div 25 = 6 $ → ✅
- $ 100 \div 50 = 2 $, $ 150 \div 50 = 3 $ → ✅

All three are valid.

---

### Final Answer

The number of positive two-digit integers that are factors of both 100 and 150 is:

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many positive two-digit integers are factors of both 100 and 150. Hmm, let me start by recalling what it means for a number to be a factor of another number. A factor of a number is an integer that divides that number without leaving a remainder. So, if a number is a factor of both 100 and 150, it must divide both of them exactly.

First, maybe I should find the common factors of 100 and 150. I remember that the common factors of two numbers are the factors of their greatest common divisor (GCD). So, if I can find the GCD of 100 and 150, then all the factors of that GCD will be the common factors of 100 and 150. That seems like a good approach. Let me try that.

To find the GCD of 100 and 150, I can use the Euclidean algorithm. Let me recall how that works. The Euclidean algorithm involves dividing the larger number by the smaller one, then replacing the larger number with the smaller number and the smaller number with the remainder from the division, and repeating this process until the remainder is zero. The last non-zero remainder is the GCD.

So, let's apply that to 100 and 150.

First, divide 150 by 100. The quotient is 1 and the remainder is 50. So, GCD(150, 100) = GCD(100, 50).

Now, divide 100 by 50. The quotient is 2 and the remainder is 0. Since the remainder is 0, the GCD is the last non-zero remainder, which is 50.

Therefore, the GCD of 100 and 150 is 50. That means all common factors of 100 and 150 are the factors of 50.

So, now I need to find all the positive factors of 50 and then determine which of those are two-digit numbers.

Let me list the factors of 50. To do that, I can start by prime factorizing 50.

50 can be broken down into prime factors as follows: 50 = 2 × 25 = 2 × 5².

The exponents of the prime factors are 1 for 2 and 2 for 5. To find the number of factors, we add 1 to each exponent and multiply them. So, (1+1)(2+1) = 2×3 = 6 factors.

But I don't need the number of factors, I need the actual factors. Let me list them.

Starting with 1, then 2, 5, 10, 25, and 50. Wait, let me check that.

Factors of 50 are numbers that divide 50 without a remainder. Let me list them systematically:

1 × 50 = 50
2 × 25 = 50
5 × 10 = 50

So, the factors are 1, 2, 5, 10, 25, 50.

So, those are all the positive factors of 50. Now, the question asks for two-digit integers. Let me check which of these factors are two-digit numbers.

Looking at the list: 1 is a single-digit number. 2 is single-digit. 5 is single-digit. 10 is two-digit. 25 is two-digit. 50 is two-digit.

So, the two-digit factors are 10, 25, and 50.

Wait, but hold on. Let me confirm that these numbers are indeed factors of both 100 and 150.

Starting with 10: 100 divided by 10 is 10, which is an integer. 150 divided by 10 is 15, which is also an integer. So, 10 is a common factor.

Next, 25: 100 divided by 25 is 4, which is an integer. 150 divided by 25 is 6, which is also an integer. So, 25 is a common factor.

Then, 50: 100 divided by 50 is 2, which is an integer. 150 divided by 50 is 3, which is also an integer. So, 50 is a common factor.

Therefore, there are three two-digit numbers that are factors of both 100 and 150.

But wait, let me make sure I didn't miss any factors. Let me check if there are any other factors of 50 that I might have missed.

Earlier, I listed 1, 2, 5, 10, 25, 50. Let me check if there are more.

Well, 50 divided by 1 is 50, which we have. Divided by 2 is 25, which we have. Divided by 5 is 10, which we have. Divided by 10 is 5, which is already listed. Divided by 25 is 2, already listed. Divided by 50 is 1, already listed. So, no other factors.

Therefore, only 10, 25, and 50 are the two-digit factors.

But hold on, let me think again. Is there a possibility that there are other common factors that aren't factors of 50? Wait, no. Because the GCD is 50, so all common factors must divide 50. Therefore, there can't be any other common factors outside of the factors of 50.

Therefore, my conclusion is that there are three two-digit integers that are factors of both 100 and 150.

But just to be thorough, let me check if there are any other numbers between 10 and 99 that divide both 100 and 150.

For example, let's take 20. Does 20 divide both 100 and 150? 100 divided by 20 is 5, which is okay. 150 divided by 20 is 7.5, which is not an integer. So, 20 is not a common factor.

How about 15? 100 divided by 15 is approximately 6.666..., which is not an integer. So, 15 is not a factor of 100.

What about 25? We already checked that.

How about 50? Also checked.

What about 10? Checked.

What about 2? No, single-digit.

What about 5? Single-digit.

What about 1? Single-digit.

How about 3? 100 divided by 3 is not an integer. 150 divided by 3 is 50, but since 3 isn't a factor of 100, it's not a common factor.

Similarly, 4? 100 divided by 4 is 25, but 150 divided by 4 is 37.5, not integer.

So, 4 isn't a common factor.

How about 6? 100 divided by 6 is not integer.

7? 100 divided by 7 is not integer.

8? 100 divided by 8 is 12.5, not integer.

9? 100 divided by 9 is not integer.

So, none of these.

How about 10, 25, 50.

Wait, what about 50? 50 is a two-digit number.

Is there another number? Let me check 25.

What about numbers like 10, 25, 50.

Is there another? Let me check 10, 25, 50.

Wait, 50 is two-digit. 25 is two-digit. 10 is two-digit.

Is there another? Let me check 50.

Wait, 50 is the largest factor.

Wait, maybe I can check the factors of 100 and 150 individually and see their common factors.

Let me try that approach as a cross-check.

First, list all factors of 100.

Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.

Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.

Now, common factors of 100 and 150 are the numbers that appear in both lists.

Looking at the factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100.

Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.

Common factors are numbers that are in both. Let me compare them:

1 is common.
2 is common.
4 is not in the factors of 150.
5 is common.
10 is common.
20 is not in factors of 150.
25 is common.
50 is common.
100 is not in factors of 150.

So, the common factors are 1, 2, 5, 10, 25, 50.

Which matches the factors of 50, as we found earlier.

Therefore, the common factors are indeed 1, 2, 5, 10, 25, 50.

So, among these, the two-digit numbers are 10, 25, 50.

Therefore, there are three such numbers.

So, my answer is 3.

But let me check once more to make sure I didn't make a mistake.

Is there a possibility that I missed a factor? For example, is there a two-digit number that divides both 100 and 150 but isn't in the list?

Looking at the common factors, they are 1, 2, 5, 10, 25, 50.

So, two-digit numbers are 10, 25, 50.

Yes, that's three numbers.

Alternatively, maybe I can think of the problem as finding the intersection of the two-digit factors of 100 and the two-digit factors of 150.

Let me try that approach.

First, find all two-digit factors of 100.

Factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100.

Two-digit factors are 10, 20, 25, 50.

Similarly, factors of 150 are 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150.

Two-digit factors are 10, 15, 25, 30, 50.

Now, the intersection of these two sets (two-digit factors of 100 and two-digit factors of 150) is {10, 25, 50}.

Therefore, again, three numbers.

So, this confirms the same result.

Therefore, the answer is 3.

But just to be absolutely thorough, let me check each of these numbers again.

10: divides 100? Yes, 100/10=10. Divides 150? 150/10=15. Yes.

25: 100/25=4, 150/25=6. Yes.

50: 100/50=2, 150/50=3. Yes.

So, all three are valid.

Is there any other number? Let me check if there are other two-digit numbers that divide both.

For example, 15: divides 150? Yes, 150/15=10. Divides 100? 100/15≈6.666... No. So, not a factor.

20: divides 100? Yes, 100/20=5. Divides 150? 150/20=7.5. No.

30: divides 150? Yes, 150/30=5. Divides 100? 100/30≈3.333... No.

So, no.

How about 5? Single-digit.

So, no.

Therefore, there are exactly three two-digit numbers.

Therefore, the answer is 3.

**Final Answer**
The number of positive two-digit integers that are factors of both 100 and 150 is \boxed{3}.

To determine how many **positive two-digit integers** are factors of **both 100 and 150**, we begin by identifying the **common factors** of these two numbers.

---

### Step 1: Find the Greatest Common Divisor (GCD)

We use the **Euclidean algorithm** to find the GCD of 100 and 150:

- $ 150 \div 100 = 1 $ remainder $ 50 $
- $ 100 \div 50 = 2 $ remainder $ 0 $

So, the GCD of 100 and 150 is **50**.

---

### Step 2: Find All Factors of the GCD

Now, we find all **positive factors** of 50:

- $ 50 = 2 \times 5^2 $
- The factors are: $ 1, 2, 5, 10, 25, 50 $

---

### Step 3: Identify Two-Digit Factors

From the list of factors of 50, we select the **two-digit numbers**:

- $ 10, 25, 50 $

These are the only two-digit numbers that divide both 100 and 150.

---

### Step 4: Verify Each Candidate

Let’s confirm that each of these numbers is indeed a factor of both 100 and 150:

- $ 100 \div 10 = 10 $, $ 150 \div 10 = 15 $ → ✅
- $ 100 \div 25 = 4 $, $ 150 \div 25 = 6 $ → ✅
- $ 100 \div 50 = 2 $, $ 150 \div 50 = 3 $ → ✅

All three are valid.

---

### Final Answer

The number of positive two-digit integers that are factors of both 100 and 150 is:

$$
\boxed{3}
$$

Solution: The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100.  Of these, the $\boxed{3}$ factors 10, 25 and 50 have two digits and are factors of 150 as well.