Track / Stage 4 / 73 of 340 #333 of 1964

Problem 333

AMC 12 late, AIME early

Geometry Difficulty 4.6 Prove it Irish Mathematical Olympiad · 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 reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.