Track / Stage 6 / 37 of 400 #1037 of 1964

Problem 1037

National Olympiad, first round

Geometry Difficulty 6.0 Prove it Local Mathematical Competitions · Romania

Let $ABCD$ be a cyclic quadrilateral. The lines $AD$ , $BC$ meet at $P$ ; $AB$ , $CD$ at $Q$ ; and $AC$ , $BD$ at $R$ . The perpendicular bisectors of $AB$ , respectively $BC$ , meet $PR$ at $X$ , respectively $QR$ at $Y$ . Prove that $XY$ passes through $B$ .

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

All poles and polars are considered with respect to the given circumcircle of $ABCD$ .
To start with, notice that line $q = PR$ is the polar of $Q$ and line $p = QR$ is the polar of $P$ . As $Y$ lies on the polar of $P$ , it follows that $P$ lies on the polar $y$ of $Y$ . The pole of the line $OY$ is the point at infinity $\infty$ on the direction $BC$ , as $OY$ is a diameter line. Thus the polar $y$ of $Y$ is the line $P\infty = BC$ , implying that $YB$ is tangent to the given circle at $B$ . Similar considerations show that $XB$ is tangent to the given circle at $B$ , hence proving the thesis.

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