Math Problem Solver

A point $D$ is chosen on side $BC$ of triangle $ABC$ with incentre $I$.

The perpendicular bisector of $BC$ meets $BI$ and $CI$ at $U$ and $V$ respectively.

The circle $(BDU)$ intersects $AB$ at $E$, and the circle $(CDV)$ intersects $AC$ at $F$.

The circles $(BDU)$ and $(CDV)$ meet at $K$.

Prove that points $E, F, K, I$ are concyclic.