Question 165
Let a triangle $ABC$ be given. Let $M$ and $N$ indicate the midpoints of the sides $AC$ and $BC$ respectively. Furthermore, let $S$ and $T$ be points respectively on the sides $AC$ and $BC$ such that:
$$
AS = \frac{1}{3} AC \quad BT = \frac{1}{3} BC.
$$
Prove that the bisectors of the angles $\angle AST$ and $\angle BTS$ meet on a point $P$ of the side $AB$ if and only if the quadrilateral $AMNB$ can be circumscribed to a circle.
$$
AS = \frac{1}{3} AC \quad BT = \frac{1}{3} BC.
$$
Prove that the bisectors of the angles $\angle AST$ and $\angle BTS$ meet on a point $P$ of the side $AB$ if and only if the quadrilateral $AMNB$ can be circumscribed to a circle.