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Annual Harvard-MIT Mathematics Tournament

Geometry Difficulty 4.6 AIME Prove it United States

Problem:

Let $ABC$ be a right triangle with hypotenuse $AC$ . Let $B'$ be the reflection of point $B$ across $AC$ , and let $C'$ be the reflection of $C$ across $AB'$ . Find the ratio of $[BCB']$ to $[BC'B']$ .

Solution

Solution:

Since $C$ , $B'$ , and $C'$ are collinear, it is evident that $[BCB'] = \frac{1}{2}[BCC']$ . It immediately follows that $[BCB'] = [BC'B']$ . Thus, the ratio is $1$ .

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