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 P1X,P2Y, P3Z are parallel, with P1X=4 and P2Y=10. From P1, draw a line parallel to XY intersecting P3Z and P2Y at Q and R, respectively. Note that P1XYR is a rectangle and that right triangles P1P2R and P1P3Q are similar. Then P3Z=P3Q+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.
