Problem:
Circles C1​ and C2​ are externally tangent, and they are both internally tangent to circle C3​. The radii of C1​ and C2​ are 4 and 10, respectively, and the centers of the three circles are collinear. A chord of C3​ is also a common external tangent of C1​ and C2​. Given that the length of the chord is mn​/p, where m,n, and p are positive integers, m and p are relatively prime, and n is not divisible by the square of any prime, find m+n+p.
Solution:
The radius of C3​ is 14. Let P1​,P2​, and P3​ be the centers of C1​,C2​, and C3​, respectively. Draw perpendiculars from P1​,P2​, and P3​ to the external tangent of C1​ and C2​ intersecting it at X,Y, and Z, respectively, so that P1​X​,P2​Y​, P3​Z​ are parallel, with P1​X=4 and P2​Y=10. From P1​, draw a line parallel to XY intersecting P3​Z​ and P2​Y​ at Q and R, respectively. Note that P1​XYR is a rectangle and that right triangles P1​P2​R and P1​P3​Q are similar. Then P3​Z=P3​Q+QZ=(10/14)⋅6+4=58/7. Because Z is the midpoint of the chord, the chord's length is 2142−(58/7)2​=8390​/7, and m+n+p=8+390+7=405​.
