Parallelogram with arbitrary segments, another parallelity needed

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Parallelogram with arbitrary segments, another parallelity needed

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Source: 2018 Latvia BW TST P12

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#1 h Mar 26, 2022, 9:28 PM

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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 $AP$ .

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Noob_at_math_69_level

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#2 h Mar 27, 2022, 10:22 AM

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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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This post has been edited 1 time. Last edited by Noob_at_math_69_level, Mar 27, 2022, 10:24 AM
Reason: hmmm

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