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Problem 857

AMC 12 late, AIME early

Geometry Difficulty 4.5 Prove it South-Afrika · South Africa · 2011

In triangle $XYZ$ , $L$ is a variable point on a fixed line passing through $X$ . $LZ$ meets $XY$ at $P$ and $LY$ meets $XZ$ at $Q$ . Show that $PQ$ passes through a fixed point on $YZ$ .

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

Let $R$ be the intersection of $YZ$ and $PQ$ . We wish to prove that $R$ is constant. Let $R'$ be the intersection of $LQ$ and $XR$ . Then it is known that $L, Y, R', Q$ form a harmonic range, i.e. $(L, Y; R', Q) = -1$ . The cross ratio depends on the angles $\angle LXY, \angle YXR', \angle R'XZ$ . Since $XL, XY$ and $XZ$ are fixed, this implies that $XR'$ is fixed. Hence $R$ is fixed since it is the intersection of $XR'$ and $YZ$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.