Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.9 AIME Find the answer

Let $P R O B L E M Z$ be a regular octagon inscribed in a circle of unit radius. Diagonals $M R, O Z$ meet at $I$ . Compute $L I$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

If $W$ is the center of the circle then $I$ is the incenter of $\triangle R W Z$ . Moreover, PRIZ is a rhombus. It follows that $P I$ is twice the inradius of a 1-1- $\sqrt{2}$ triangle, hence the answer of $2-\sqrt{2}$ . So $L I=\sqrt{2}$ . Alternatively, one can show (note, really) that the triangle $O I L$ is isosceles.

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