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equal sums of triangle areas, a+c = b+d, midpoints related

Source: Norwegian Mathematical Olympiad 1993 - Abel Competition p1a

February 11, 2020
geometrytriangle areaareamidpoints

Problem Statement

Let ABCDABCD be a convex quadrilateral and A′,B′C′,D′A',B'C',D' be the midpoints of AB,BC,CD,DAAB,BC,CD,DA, respectively. Let a,b,c,da,b,c,d denote the areas of quadrilaterals into which lines A′C′A'C' and B′D′B'D' divide the quadrilateral ABCDABCD (where a corresponds to vertex AA etc.). Prove that a+c=b+da+c = b+d.
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