is a - - triangle in which three congruent circles are each tangent to two sides of the triangle and are concurrent at . What is their radius? Express it as , where and are coprime positive integers and submit .
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Let D , E , and F be the centers of the three congruent circles with radius r , and draw in and label the diagram as follows:
By Heron's formula, the area of △ A B C is A △ A B C = s ( s − a ) ( s − b ) ( s − c ) = 2 1 ( 2 1 − 1 3 ) ( 2 1 − 1 4 ) ( 2 1 − 1 5 ) = 8 4 .
Since each circle is tangent to the sides of △ A B C , the centers D , E , and F are on the angle bisectors of A , B , and C .
A 1 3 - 1 4 - 1 5 triangle is a 5 - 1 2 - 1 3 triangle combined with a 9 - 1 2 - 1 5 , so A G = D G cot ∠ D A G = r cot ( 2 1 ∠ C A B ) = r cot ( 2 1 cos − 1 ( 1 3 5 ) ) = 2 3 r and H B = E H cot ∠ E B H = r cot ( 2 1 ∠ A B C ) = r cot ( 2 1 cos − 1 ( 1 5 9 ) ) = 2 r , so that D E = G H = A B − A G − H B = 1 4 − 2 3 r − 2 r = 1 4 − 2 7 r .
Since the sides of △ D E F are parallel to the sides of △ A B C , the two triangles are similar, and have a side ratio of A B D E = 1 4 1 4 − 2 7 r = 1 − 4 1 r .
Since Z D = Z E = Z F = r , Z is the circumcenter of △ D E F , so 2 r = 2 A △ D E F D E ⋅ E F ⋅ D F = ( 1 − 4 1 r ) 2 A △ A B C ( 1 − 4 1 r ) A B ⋅ ( 1 − 4 1 r ) B C ⋅ ( 1 − 4 1 r ) A C = 8 4 1 4 ⋅ 1 5 ⋅ 1 3 ⋅ ( 1 − 4 1 r ) , and 2 r = 8 4 1 4 ⋅ 1 5 ⋅ 1 3 ⋅ ( 1 − 4 1 r ) solves to r = 9 7 2 6 0 .
Therefore, p = 2 6 0 , q = 9 7 , and p + q = 3 5 7 .