HMMT February

Geometry Difficulty 4.9 AIME Find the answer

A circle inscribed in a square has two chords as shown in a pair. It has radius 2, and $P$ bisects $T U$ . The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle.

Your answer

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

Solution

The point lies between $X$ and $Q$ . Then $M N X Q$ is a parallelogram. For, $O B \| N M$ by homothety at $C$ and $P M \| N X$ because $M N X P$ is an isoceles trapezoid. It follows that $Q X=M N$ . Considering that the center of the circle together with points $M, C$ , and $N$ determines a square of side length 2, it follows that $M N=2 \sqrt{2}$ , so the answer is $2 \sqrt{2}-2$ .

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