AoPS - Problem

Processing math: 100%

Problem

Source: 2018 Latvia BW TST P12

Tags: geometry, parallelogram

Blastoor

27.03.2022 00:28

Let $ABCD$ be a parallelogram. Let $X$ and $Y$ be arbitrary points on sides $BC$ and $CD$ , respectively. Segments $BY$ and $DX$ intersect at $P$ . Prove that the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$ .

Noob_at_math_69_level

27.03.2022 13:22

Let $M$ , $N$ , $Q$ be mid points of segments $BD$ , $XY$ , $CP$ . Since $MN$ is the Gauss line of complete quadrilateral $YCXP.DB$ so $M,N,Q$ are collinear. Since $ABCD$ is a parallelogram so $M$ is also mid point of $CA$ .Since $M$ and $Q$ are mid points of $CA$ and $CP$ so $MQ//AP$ but $M,N,Q$ are collinear so $MN//AP$ so the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$ .

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