Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Find the answer

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 $V$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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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