A triangle $(ABC)$ and a point $D$ in its plane satisfy the relations \[\frac{BC}{AD}=\frac{CA}{BD}=\frac{AB}{CD}=\sqrt{3}.\] Prove that $(ABC)$ is equilateral and $D$ is its center.
Source: IMO 1980 GB-5
Tags: geometry unsolved, geometry
A triangle $(ABC)$ and a point $D$ in its plane satisfy the relations \[\frac{BC}{AD}=\frac{CA}{BD}=\frac{AB}{CD}=\sqrt{3}.\] Prove that $(ABC)$ is equilateral and $D$ is its center.