Library / Romanian Mathematical Olympiad /8 of 275

Romanian Mathematical Olympiad

Geometry Difficulty 4.8 AIME Prove it Romania

Let $ABCD$ be a convex quadrilateral in which the diagonals intersect at $O$ . Given that $\overrightarrow{AB} + \overrightarrow{AD} + \overrightarrow{AO} = \overrightarrow{BC} + \overrightarrow{DC} + \overrightarrow{OC}$ , prove that $ABCD$ is a parallelogram.

Marius Dolcan

Solution

We obtain that the points $M$ , $N$ and $O$ are collinear, so $M = N = O$ . Thus point $O$ is the midpoint of each of the diagonals, therefore $ABCD$ is a parallelogram.

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.