Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Prove it Bulgaria

Determine the locus of the points $M$ which are interior for the convex quadrilateral $ABCD$ and the areas of the quadrilaterals $ABCM$ and $AMCD$ are equal.

Solution

The locus is the segment $PQ$ such that $PQ \parallel AC$ and $d(PQ, AC) = \frac{|S_{ABC} - S_{ACD}|}{AC}$ , where $S_{ABC}$ and $S_{ACD}$ are the areas of triangles $ABC$ and $ACD$ , respectively.

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