Problem

Source: 2019 Slovenia 2nd TST Problem 3

Tags: Team Selection Test, geometry



Let $ABC$ be a non-right triangle and let $M$ be the midpoint of $BC$. Let $D$ be a point on $AM$ (D≠A, D≠M). Let ω1 be a circle through $D$ that intersects $BC$ at $B$ and let ω2 be a circle through $D$ that intersects $BC$ at $C$. Let $AB$ intersect ω1 at $B$ and $E$, and let $AC$ intersect ω2 at $C$ and $F$. Prove, that the tangent on ω1 at $E$ and the tangent on ω2 at $F$ intersect on $AM$.