Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Prove it Soviet Union

Problem:

The line joining the midpoints of two opposite sides of a convex quadrilateral makes equal angles with the diagonals. Show that the diagonals are equal.

Solution

Solution:

Let $L$ , $M$ , $N$ be the midpoints of $BC$ , $CD$ , $DA$ . Assume that $NL$ makes equal angles with $AC$ and $BD$ , so $\angle NLM = \angle BEL = \angle AFN = \angle LNM$ , so $LM = MN$ and hence $BD = AC$ .

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