Baltic Way 2019 · Problem 12

Geometry

Let $ABC$ be a triangle and $H$ its orthocenter. Let $D$ be a point lying on the segment $AC$ and let $E$ be the point on the line $BC$ such that $BC\perp DE$ . Prove that $EH\perp BD$ if and only if $BD$ bisects $AE$ .

Change pool

When you’re ready

Review material becomes available with the next Daily.

Review

Topics

Angles and distances · Triangles and centers

Solutions

Solution

Let $BD \cap AH = X$ . Then $XH \perp BE$ , so $EH \perp BD \iff H$ is orthocenter $BXE \iff BH \perp EX \iff EX \parallel AC \iff AXED$ is parallelogram $\iff BD$ bisects $AE$ .

Contest context

Results from Baltic Way 2019

11 teams

Mean score: 3.7 / 5
Scores of 4 or 5: 8 / 11
Estonia: 5 / 5

Score distribution

All team scores

Team	Score
St. Petersburg	5 / 5
Poland	5 / 5
Estonia	5 / 5
Lithuania	5 / 5
Germany	1 / 5
Norway	5 / 5
Finland	0 / 5
Denmark	5 / 5
Sweden	5 / 5
Latvia	5 / 5
Iceland	0 / 5

Elsewhere

AoPS · Baltic Way 2019
Official results