Berkeley Math Circle

Geometry Difficulty 4.2 AIME Prove it United States

Problem:

Let $A B C D E F$ be a convex hexagon such that the quadrilaterals $A B D E$ and $A C D F$ are parallelograms. Prove that $B C E F$ is also a parallelogram.

Solution

Solution:

It is well known that a quadrilateral is a parallelogram if and only if the two diagonals bisect each other, that is, have a common midpoint. Since $A B D E$ is a parallelogram, the midpoints of $A D$ and $B E$ coincide; since $A C D F$ is a parallelogram, the midpoints of $A D$ and $C F$ coincide. Consequently the midpoints of $B E$ and $C F$ coincide and $B C E F$ is a parallelogram.

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