HMMT 二月 2007 · 冲刺赛 · 第 8 题

HMMT February 2007 — Guts Round — Problem 8

[ 7 ] 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.

解析

[ 7 ] 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. √ Answer: 2 2 − 2 . Let OB intersect the circle at X and Y , and the chord P M at Q , such that O lies between X and Q . Then M N XQ is a parallelogram. For, OB ‖ N M by homothety at C and P M ‖ N X because M N XP is an isoceles trapezoid. It follows that QX = 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 2, so the answer is 2 2 − 2 .