Problem:
Circles C1​ and C2​ each have radius 1 , and the distance between their centers is 21​. Circle C3​ is the largest circle internally tangent to both C1​ and C2​. Circle C4​ is internally tangent to both C1​ and C2​ and externally tangent to C3​. What is the radius of C4​?
Answer Choices:
A. 141​
B. 121​
C. 101​
D. 283​
E. 91​
Solution:
Let r be the radius of C4​. Let the point Pk​ be the center of Ck​ for 1≤k≤4. Let A be the point of intersection of circles C1​ and C4​ and let ℓ be the tangent line to P1​ at A. By symmetry, P3​ is the midpoint of P1​P2​​, and P1​P3​=41​. Also by symmetry, P4​ lies on the perpendicular bisector of P1​P2​​. The radius P1​A​ is perpendicular to ℓ. Likewise the radius P4​A​ is perpendicular to ℓ, so P1​,P4​, and A all lie on the segment P1​A​ with P4​ between P1​ and A. Then P1​P4​=1−r.
The radius of C3​ is 43​. Then P3​P4​=43​+r, and △P1​P3​P4​ is a right triangle with P1​P3​=41​, P3​P4​=43​+r, and P1​P4​=1−r. By the Pythagorean Theorem,
(41​)2+(43​+r)2=(1−r)2
Expanding gives
161​+169​+23r​+r2=1−2r+r2
which gives r=(D)283​​.
The problems on this page are the property of the MAA's American Mathematics Competitions
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