Track / Stage 4 / 53 of 340 #313 of 1964

Problem 313

AMC 12 late, AIME early

Geometry Difficulty 4.6 Prove it IRL_ABooklet · Ireland

Let $A$ , $B$ , $C$ be three points on a circle of centre $O$ . The perpendicular line from $O$ to $BC$ intersects line $AC$ at $P$ , and the perpendicular line from $O$ to $AC$ intersects line $BC$ at $Q$ . Let $L$ be the midpoint of $OC$ and $K$ the midpoint of $PQ$ .
Prove that $KL$ is perpendicular on $AB$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

The quadrilateral *CNOM* is cyclic because $\angle CNO = 90^\circ$ and $\angle CMO = 90^\circ$ , and $OC$ is a diameter and $L$ the centre of its circumcircle.

The quadrilateral *MNPQ* is cyclic as well since $\angle PNQ = \angle PMQ = 90^\circ$ , and $PQ$ is a diameter and $K$ the centre of its circumcircle. The line *MN* is the radical axis of these two circles and so is perpendicular to the line *KL* which connects the centres of the circles. Because *M* and *N* are midpoints of sides of $\triangle ABC$ , $MN \parallel AB$ , hence $AB \perp KL$ .

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.