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Problem 1061

AMC 12 late, AIME early

Geometry Difficulty 4.7 Prove it Hong Kong competition problems · Hong Kong

Given triangle $ABC$ , let $D$ be an inner point of the segment $BC$ . Let $P$ and $Q$ be distinct inner points of the segment $AD$ . Let $K = BP \cap AC$ , $L = CP \cap AB$ , $E = BQ \cap AC$ , $F = CQ \cap AB$ . Given that $KL \parallel EF$ , find all possible values of the ratio $BD : DC$ .

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 only possible value is $1$ .
We use projective geometry. Consider the following projection.
$AB(A, L, F, B) \xrightarrow{\{C\}} AD(A, P, Q, D) \xrightarrow{\{B\}} AC(A, K, E, C).$
This shows $(A, L, F, B)$ and $(A, K, E, C)$ are perspective, and so $LK$ , $FE$ , $BC$ are concurrent. Since $KL \parallel EF$ , this shows $KL \parallel EF \parallel CB$ .
Now, since $AD$ , $BK$ , $CL$ are concurrent, $KL \cap CB$ is the harmonic conjugate of $D$ with respect to $BC$ . As $KL \cap CB$ is a point at infinity, $D$ must be the midpoint of $BC$ .

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