Library / All-Soviet-Union Mathematical Olympiad /24 of 185

All-Soviet-Union Mathematical Olympiad

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 $BD = AC$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.