Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 8.2 Shortlist Prove it Hong Kong

Let $ABCD$ be a convex quadrilateral. $E$ and $F$ are points on the diagonal $AC$ such that $E$ and $F$ are interior points of triangles $ABD$ and $BCD$ respectively. Suppose $BE$ extend cuts $AD$ at $P$ , $DE$ extend cuts $AB$ at $Q$ , $DF$ extend cuts $BC$ at $R$ and $BF$ extend cuts $DC$ at $S$ . Show that the three lines $QP$ , $BD$ and $RS$ are either parallel or concurrent.

Solution

Let $Y$ be the intersection point of $AC$ and $BD$ . Since $AY$ , $BP$ , $DQ$ are concurrent, $QP$ meets $BD$ at the harmonic conjugate $X$ of $Y$ with respect to $B$ , $D$ (note that $X$ could be a point at infinity). Similarly, since $CY$ , $BS$ , $DR$ are concurrent, $RS$ meets $BD$ at $X$ . This shows $QP$ , $BD$ , $RS$ are concurrent in the projective sense. This means they are concurrent or parallel.

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