Problem:
A circle of radius 1 is tangent to a circle of radius 2. The sides of △ABC are tangent to the circles as shown, and the sides AB and AC are congruent. What is the area of △ABC?
Answer Choices:
A. 235
B. 152
C. 364
D. 162
E. 24
Solution:
Let O and O′ denote the centers of the smaller and larger circles, respectively. Let D and D′ be the points on AC that are also on the smaller and larger circles, respectively. Since △ADO and △AD′O′ are similar right triangles, we have
1AO=2AO′=2AO+3, so AO=3
As a consequence,
AD=AO2−OD2=9−1=22
Let F be the midpoint of BC. Since △ADO and △AFC are similar right triangles, we have
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