Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 6.0 National olympiad Prove it 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 $B$ .

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.

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