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Problem 908

AMC 12 late, AIME early

Geometry Difficulty 4.0 Prove it BMO Round 1 · United Kingdom · 2011

Let $ABC$ be a triangle with $\angle CAB$ a right-angle. The point $L$ lies on the side $BC$ between $B$ and $C$ . The circle $ABL$ meets the line $AC$ again at $M$ and the circle $CAL$ meets the line $AB$ again at $N$ . Prove that $L$ , $M$ and $N$ lie on a straight line.

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

We don't reproduce this publisher's solutions. Their own solution is here — work the problem first.

Source: UK Mathematics Trust, licensed © UK Mathematics Trust; question papers published free at bmos.ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.