Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Prove it 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 $AB$ .

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 $AB \perp KL$ .

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