<Question>
In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown. [asy]
size(200);
pair A, B, C, P, Q, R, S;
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);
draw(A--B--C--A);
draw(circle(P, 1));
draw(circle(Q, 1));
draw(circle(R, 1));
draw(circle(S, 1));
label("A", A, N);
label("B", B, SW);
label("C", C, SE);
dot(P);
dot(Q);
dot(R);
dot(S);
label("P", P, N);
label("Q", Q, SW);
label("R", R, SW);
label("S", S, SE);
[/asy]


What is the degree measure of the smallest angle in triangle $PQS$?
</Question>

<think>
<Plan_1>
Okay, so I need to find the smallest angle in triangle PQS. Let me start by understanding the diagram and the given information. There are four circles with radius 1, centered at P, Q, R, and S. They are all tangent to each other and to the sides of triangle ABC. The Asymptote code gives some coordinates, which might help.

<prompt_1.1> Analyze the given coordinates of the centers P, Q, R, and S and their implications for the triangle's geometry. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, let me note down the coordinates provided in the Asymptote code:

- R is at (0, 0)
- Q is at (-2, 0)
- S is at (2, 0)
- P is at (1, 1.732)

So triangle ABC is drawn with vertices A, B, and C. The circles are centered at these points, each with radius 1. The circles at P, Q, R, S are all tangent to each other and to the sides of the triangle. 

Since all circles are tangent to each other, the distance between any two centers should be equal to the sum of their radii. But since all radii are 1, the distance between any two centers should be 2. Let me verify that.

First, between Q and R: Q is at (-2, 0), R is at (0, 0). The distance between them is sqrt[(0 - (-2))^2 + (0 - 0)^2] = sqrt[4] = 2. That checks out.

Between R and S: R (0,0) to S (2,0) is also 2 units. Similarly, Q to S is from (-2,0) to (2,0), which is 4 units. Wait, but if all circles are tangent to each other, then the distance between Q and S should be 2 (since both have radius 1). But the distance here is 4. That seems contradictory. Hmm. Maybe I misunderstood the problem. Let me read again.

The problem states: "four circles of radius 1 with centres P, Q, R, and S are tangent to one another and to the sides of triangle ABC". Hmm, "tangent to one another" – does that mean each circle is tangent to every other circle? That would imply that all centers are 2 units apart. But in the given coordinates, Q is at (-2,0), S is at (2,0), so distance QS is 4. That can't be. Therefore, perhaps "tangent to one another" doesn't mean all pairs, but that they are arranged in a chain or some specific tangency. Wait, maybe the circles are tangent to their neighboring circles. For example, P is tangent to Q and R, Q is tangent to P and R, R is tangent to P, Q, S, and S is tangent to R. But in that case, the distance between Q and S would still be 4, which is more than 2. So perhaps the circles are not all mutually tangent, but each is tangent to its adjacent ones?

Alternatively, maybe the circles are arranged such that each is tangent to two others, forming a sort of quadrilateral. But given the coordinates, let's see:

If Q is at (-2,0), R is at (0,0), S is at (2,0), and P is at (1, 1.732). So Q, R, S are colinear along the x-axis, each 2 units apart. Then P is above R, at (1, 1.732). Let's compute the distances:

- PQ: distance between (-2,0) and (1, 1.732). Let's compute that.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
sqrt[(1 - (-2))² + (1.732 - 0)²] = sqrt[9 + 3] = sqrt[12] ≈ 3.464. But if the circles at P and Q both have radius 1, then the distance between centers should be 2. But here it's sqrt(12).

<prompt_2.1> Verify the distances between the centers of the circles and identify any contradictions with the given radius. </prompt_2.1>

Wait, perhaps the Asymptote code's coordinates are not to scale? Because if all circles are radius 1 and tangent to each other, the centers should form a square or something. But in the given coordinates, the horizontal distance between Q and S is 4, which would require their centers to be 4 units apart, but with radius 1 each, the distance should be 2. So there's a discrepancy here.

Alternatively, maybe the circles are not all tangent to each other, but each is tangent to its neighbors. For example, Q is tangent to R, R is tangent to S, and P is tangent to R and maybe Q and S? Let me check.

<prompt_2.2> Re-evaluate the tangency conditions and verify the distances between the centers. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
That doesn't make sense. So there must be a mistake here. Maybe the coordinates in the Asymptote code are not to scale? Or perhaps I need to re-examine the problem.

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 all tangent to each other. That would mean that each pair of circles is tangent. But with four circles, each pair would need to be tangent, which would require a complete graph of tangencies. However, in 2D, arranging four circles all tangent to each other is more complex. The usual case is three circles mutually tangent, forming a sort of triangle. Adding a fourth circle tangent to all three would require it to be in a specific position. But in this case, the centers are given as P, Q, R, S with specific coordinates. 

Alternatively, maybe "tangent to one another" doesn't mean all pairs, but that they are tangent in some way, perhaps each adjacent pair. But the wording is a bit ambiguous. However, given the coordinates, let's proceed with the given data.
</execution_2.1>

<execution_2.2>
Distance from P to Q: sqrt[(1 - (-2))² + (1.732 - 0)²] = sqrt[9 + 3] = sqrt[12] ≈ 3.464. If both have radius 1, then the distance should be 2. So that's not possible. Therefore, perhaps the Asymptote code's coordinates are illustrative but not to scale. Maybe the actual problem is constructed such that the centers form a specific triangle, and we need to figure out the angles based on the tangency conditions.

Alternatively, maybe the triangle ABC is such that the circles are arranged with R at the origin, Q to the left, S to the right, and P above R. The circles at Q, R, S are along the base BC, each tangent to the base and to each other. Then the circle at P is tangent to the two circles above? Wait, maybe not. Let me try to visualize.

Given that all circles are tangent to the sides of triangle ABC, which is a larger triangle. The centers P, Q, R, S are inside ABC. The circles at Q, R, S are likely tangent to the base BC, since their centers are along the x-axis (assuming R is at (0,0), Q at (-2,0), S at (2,0)). Then the circle at P is above R, at (1, 1.732). The distance from P to R is sqrt[(1-0)^2 + (1.732 - 0)^2] = sqrt[1 + 3] = sqrt[4] = 2. So that's correct, since both circles have radius 1, so distance between centers should be 2. Similarly, distance from P to Q: sqrt[(1 - (-2))^2 + (1.732 - 0)^2] = sqrt[9 + 3] = sqrt[12] ≈ 3.464. But that's more than 2, which would mean the circles are not tangent. However, the problem states all four circles are tangent to one another. Hmm, maybe there's a mistake here.

Wait, perhaps the circle at P is only tangent to the circle at R, but not to Q or S? But the problem says "tangent to one another," which could mean each pair of circles is tangent. But in that case, the distance between P and Q, P and S, Q and S should all be 2. But according to coordinates, that's not the case. So maybe the coordinates are just a rough sketch, and we need to figure out the actual positions based on the problem statement.

Alternatively, perhaps the centers form a rhombus? Let's see: Q, R, S are on a line, each 2 units apart. Then P is above R. If PR is 2 units (since radius 1 each, so centers 2 units apart), then P is at (1, sqrt(3)), since the distance from R (0,0) to P (1, sqrt(3)) is 2. Because sqrt(1^2 + (sqrt(3))^2) = sqrt(1 + 3) = 2. So that makes sense. Then, the coordinates would be:

- Q: (-2, 0)
- R: (0, 0)
- S: (2, 0)
- P: (1, sqrt(3)) ≈ (1, 1.732)

So that's a rhombus with sides QR = 2, PR = 2, but the distances PQ and PS would be sqrt[(1 - (-2))^2 + (sqrt(3) - 0)^2] = sqrt[9 + 3] = sqrt(12) ≈ 3.464. Similarly, PS is the same. But if the circles at P and Q have centers 3.464 units apart, but both have radius 1, they can't be tangent. So that's a problem.

Wait, maybe the problem is that the circles are tangent to each other and also tangent to the sides of the triangle. So maybe the circles at Q, R, S are each tangent to two sides of the triangle ABC, and the circle at P is tangent to the other two sides? Let me think.

If ABC is a triangle with a circle at each corner (Q, R, S) tangent to two sides, and another circle (P) tangent to the other two sides and to the corner circles. That might form a configuration where the centers are arranged such that the circles are tangent to each other and the sides. But in that case, the distances between centers would be 2 (sum of radii, which is 1 + 1). So perhaps the coordinates provided in the Asymptote code are not accurate, or perhaps they are scaled.

Alternatively, maybe the triangle ABC is equilateral? Given that P is at (1, sqrt(3)), which is the height of an equilateral triangle with side length 2. But in the Asymptote code, ABC has coordinates A(1.366, 3.098), B(-5.73,-1), C(3.732,-1). That seems like a different triangle. Maybe it's a 30-60-90 triangle?

Alternatively, perhaps the centers form a triangle PQS, and we need to find the angles in that triangle. The problem asks for the smallest angle in triangle PQS. So maybe I need to compute the lengths of the sides PQ, QS, and PS, then use the Law of Cosines to find the angles.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.2:
First, let's get the coordinates straight.

From the Asymptote code:

<prompt_3.1> Calculate the distances between the centers P, Q, and S to determine the side lengths of triangle PQS. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
- R is at (0, 0)
- Q is at (-2, 0)
- S is at (2, 0)
- P is at (1, 1.732)

So:

Coordinates:
- P: (1, 1.732)
- Q: (-2, 0)
- S: (2, 0)

First, compute the lengths of the sides of triangle PQS.

1. Length PQ: distance between P(1, 1.732) and Q(-2, 0)
Difference in x: 1 - (-2) = 3
Difference in y: 1.732 - 0 = 1.732
So PQ = sqrt(3² + (1.732)²) = sqrt(9 + 3) = sqrt(12) ≈ 3.464

2. Length PS: distance between P(1, 1.732) and S(2, 0)
Difference in x: 1 - 2 = -1
Difference in y: 1.732 - 0 = 1.732
So PS = sqrt((-1)² + (1.732)²) = sqrt(1 + 3) = sqrt(4) = 2

3. Length QS: distance between Q(-2, 0) and S(2, 0)
Difference in x: -2 - 2 = -4
Difference in y: 0 - 0 = 0
So QS = sqrt((-4)² + 0²) = sqrt(16) = 4

Therefore, triangle PQS has sides of length 2, sqrt(12), and 4.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Wait, sqrt(12) is approximately 3.464, so sides are 2, 3.464, 4.

Now, to find the angles of triangle PQS. The smallest angle will be opposite the shortest side, which is PQ? Wait, no: the sides are PQ = sqrt(12), PS = 2, QS = 4. Wait, hold on:

<prompt_4.1> Identify the shortest side in triangle PQS and determine the smallest angle. </prompt_4.1>

But let's confirm using the Law of Cosines.

First, let's label the triangle:

<prompt_4.2> Use the Law of Cosines to calculate the angles of triangle PQS. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
Wait, sides:

- PQ: sqrt(12) ≈ 3.464
- PS: 2
- QS: 4

So the sides in order are PS (2), PQ (≈3.464), QS (4). Therefore, the shortest side is PS, which is 2 units. Therefore, the smallest angle is opposite PS, which is angle Q. Because in triangle PQS, side PS is opposite angle Q. Wait, let's confirm:

In triangle PQS:

- Side PQ is opposite angle S
- Side PS is opposite angle Q
- Side QS is opposite angle P

Wait, no. Wait, in triangle PQS, the vertices are P, Q, S. So:

- Angle at P: between sides PQ and PS, opposite side QS
- Angle at Q: between sides PQ and QS, opposite side PS
- Angle at S: between sides PS and QS, opposite side PQ

Therefore, the smallest side is PS (2 units), which is opposite angle Q. Therefore, the smallest angle is angle Q.
</execution_4.1>

<execution_4.2>
- Let’s call point P, Q, S.

- Sides:

- PQ: between P and Q: sqrt(12) ≈ 3.464

- PS: between P and S: 2

- QS: between Q and S: 4

Therefore, the sides are:

- Opposite angle P: QS = 4

- Opposite angle Q: PS = 2

- Opposite angle S: PQ = sqrt(12)

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

Wait, but let me verify.

Wait, in triangle PQS:

- Vertex P is connected to Q and S. So side QS is opposite angle P.

- Vertex Q is connected to P and S. So side PS is opposite angle Q.

- Vertex S is connected to P and Q. So side PQ is opposite angle S.

Therefore, the sides and opposite angles:

- PS (2) opposite angle Q

- PQ (sqrt(12)) opposite angle S

- QS (4) opposite angle P

Therefore, since PS is the shortest side (2), the angle opposite to it, which is angle Q, is the smallest angle. Therefore, we need to compute angle Q and find its measure.

Alternatively, maybe angle S is opposite side PQ (sqrt(12)), which is longer, so angle S is larger. So angle Q is the smallest.
</execution_4.2>
</Execution_4>

<Plan_5>
Based on execution_4.2:
To compute angle Q, we can use the Law of Cosines. In triangle PQS, angle Q is between sides PQ and QS, and opposite side PS.

Law of Cosines formula:

<prompt_5.1> Use the Law of Cosines to calculate angle Q in triangle PQS. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
c² = a² + b² - 2ab cos(C)

Here, to find angle Q, which is at vertex Q. The sides adjacent to angle Q are PQ and QS, and the side opposite is PS.

So:

PS² = PQ² + QS² - 2 * PQ * QS * cos(angle Q)

Plugging in the values:

2² = (sqrt(12))² + 4² - 2 * sqrt(12) * 4 * cos(angle Q)

Compute each term:

4 = 12 + 16 - 8 * sqrt(12) * cos(angle Q)

Simplify:

4 = 28 - 8 * sqrt(12) * cos(angle Q)

Subtract 28 from both sides:

4 - 28 = -8 * sqrt(12) * cos(angle Q)

-24 = -8 * sqrt(12) * cos(angle Q)

Divide both sides by -8 * sqrt(12):

(-24) / (-8 * sqrt(12)) = cos(angle Q)

Simplify:

24 / (8 * sqrt(12)) = cos(angle Q)

3 / sqrt(12) = cos(angle Q)

Simplify sqrt(12) as 2 * sqrt(3):

3 / (2 * sqrt(3)) = cos(angle Q)

Multiply numerator and denominator by sqrt(3):

(3 * sqrt(3)) / (2 * 3) = cos(angle Q)

Simplify:

sqrt(3)/2 = cos(angle Q)

Therefore, angle Q is arccos(sqrt(3)/2). What angle has cosine sqrt(3)/2? That's 30 degrees. Because cos(30°) = sqrt(3)/2. Therefore, angle Q is 30 degrees.

Therefore, the smallest angle in triangle PQS is 30 degrees.
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
But let me cross-verify using another method. For example, maybe using the Law of Sines.

<prompt_6.1> Cross-verify the smallest angle using the Law of Sines in triangle PQS. </prompt_6.1>
</Plan_6>

<Execution_6>
<execution_6.1>
First, let's compute all angles.

We already found angle Q is 30 degrees. Let's check angle S. Side PQ is sqrt(12), which is opposite angle S. Using Law of Sines:

sin(angle S) / PQ = sin(angle Q) / PS

So:

sin(angle S) / sqrt(12) = sin(30°) / 2

sin(angle S) = (sqrt(12) * sin(30°)) / 2

sin(30°) = 0.5, so:

sin(angle S) = (sqrt(12) * 0.5) / 2 = sqrt(12) / 4 ≈ 3.464 / 4 ≈ 0.866

So sin(angle S) ≈ 0.866. Therefore, angle S ≈ arcsin(0.866) ≈ 60 degrees.

Similarly, angle P can be found since the sum of angles in a triangle is 180°. So angle P = 180 - 30 - 60 = 90 degrees.

Therefore, angles are 30°, 60°, 90°, which sum to 180°. Therefore, the smallest angle is indeed 30°, which is angle Q. Therefore, the answer is 30 degrees.
</execution_6.1>
</Execution_6>

<Plan_7>
Based on execution_6.1:
But let me check again if I didn't make a mistake in the Law of Cosines step.

<prompt_7.1> Verify the Law of Cosines calculation for angle Q in triangle PQS. </prompt_7.1>
</Plan_7>

<Execution_7>
<execution_7.1>
We had:

PS² = PQ² + QS² - 2 * PQ * QS * cos(angle Q)

4 = 12 + 16 - 2 * sqrt(12) * 4 * cos(angle Q)

Wait, hold on, wait. Wait, PQ is sqrt(12), so PQ squared is 12. QS is 4, so squared is 16. So 12 + 16 = 28. Then 2 * PQ * QS is 2 * sqrt(12) * 4 = 8 * sqrt(12). Then:

4 = 28 - 8 * sqrt(12) * cos(angle Q)

So 8 * sqrt(12) * cos(angle Q) = 28 - 4 = 24

Therefore, cos(angle Q) = 24 / (8 * sqrt(12)) = 3 / sqrt(12) = sqrt(3)/2. Yes, that's correct.

Therefore, angle Q is 30 degrees. Therefore, the smallest angle is 30 degrees.

But just to make sure, let's think about the triangle. If the sides are 2, 4, and 2*sqrt(3) ≈ 3.464. Wait, but in our case, the sides are 2, sqrt(12) ≈ 3.464, and 4. So the triangle has sides 2, 3.464, 4. So the sides are in the ratio 2 : 3.464 : 4. The angles opposite these sides would be increasing with the side lengths. So the smallest angle is opposite the shortest side, which is 2, which is angle Q.
</execution_7.1>
</Execution_7>

<Plan_8>
Based on execution_7.1:
Therefore, the degree measure of the smallest angle in triangle PQS is 30 degrees.
**Final Answer**
The degree measure of the smallest angle in triangle \(PQS\) is \boxed{30}.
</Plan_8>

<Execution_8>
No prompt to execute.
</Execution_8>

<Final_answer>
Therefore, the degree measure of the smallest angle in triangle $PQS$ is $\boxed{30^\circ}$.
</Final_answer>
</think>