HMMT November

Geometry Difficulty 4.9 AIME Prove it United States

Problem:

Let $L O V E R$ be a convex pentagon such that $L O V E$ is a rectangle. Given that $O V = 20$ and $L O = V E = R E = R L = 23$ , compute the radius of the circle passing through $R$ , $O$ , and $V$ .

Solution

Solution:

Let $X$ be the point such that $R X O L$ is a rhombus. Note that line $R X$ defines a line of symmetry on the pentagon $L O V E R$ . Then by symmetry $R X V E$ is also a rhombus, so $R X = O X = V X = 23$ . This makes $X$ the center of the circle, and the radius is $23$ .

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