Problem

Source: 2006 Romania JBMO TST1 p1

Tags: equal ratio, geometry, angle, ratio, right triangle



Let $ABC$ be a triangle right in $C$ and the points $D, E$ on the sides $BC$ and $CA$ respectively, such that $\frac{BD}{AC} =\frac{AE}{CD} = k$. Lines $BE$ and $AD$ intersect at $O$. Show that the angle $\angle BOD = 60^o$ if and only if $k =\sqrt3$.