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XII OBM

Brazil geometry

Problem

is a convex quadrilateral. are the midpoints of sides respectively. Find the point such that area .
Solution
Let be the midpoint of , the midpoint of . Take so that is parallel to and is parallel to . Now is a midline of , so . Similarly, . So . But is parallel to , so . Hence



Similarly for and the other quadrilaterals.
Final answer
Let M and N be the midpoints of AC and BD. Take P as the intersection of the line through M parallel to BD and the line through N parallel to AC. For this P, the four areas are equal (each equals one quarter of the area of ABCD).

Techniques

Constructions and loci