Let $ABCD$ be a quadrilateral whose diagonals are not perpendicular and whose sides $AB$ and $CD$ are not parallel.Let $O$ be the intersection of its diagonals.Denote with $H_1$ and $H_2$ the orthocenters of triangles $AOB$ and $COD,$ respectively.If $M$ and $N$ are the midpoints of the segment lines $AB$ and $CD,$ respectively.Prove that the lines $H_1H_2$ and $MN$ are parallel if and only if $AC=BD.$
Problem
Source: JBMO Shortlist 2014,G6
Tags: geometry
MilosMilicev
06.11.2016 17:12
Radical axis!
den_thewhitelion
06.11.2016 17:21
https://www.artofproblemsolving.com/community/c6h1089039p4831853
MazeaLarius
21.03.2018 14:43
Complex numbers.
Drunken_Master
21.03.2018 16:20
Orkhan-Ashraf_2002 wrote: Let $ABCD$ be a quadrilateral whose diagonals are not perpendicular and whose sides $AB$ and $CD$ are not parallel.Let $O$ be the intersection of its diagonals.Denote with $H_1$ and $H_2$ the orthocenters of triangles $AOB$ and $COD,$ respectively.If $M$ and $N$ are the midpoints of the segment lines $AB$ and $CD,$ respectively.Prove that the lines $H_1H_2$ and $MN$ are parallel if and only if $AC=BD.$ $H_1H_2$ is the steiner line of complete quadrilateral $ABDC$. If $P,Q$ are midpoints of $AD,BC;$ then $PQ$ is the gauss line and thus $H_1H_2 \perp PQ$. Now, $MN \parallel H_1H_2 \iff MN \perp PQ$. Since $MNPQ$ is always a parallelogram, $MN \perp PQ \iff MNPQ$ is a rhombus, which happens iff $AC=BD$. $\square$