Plane Geometry 2

[rsoldo]

A circle is inscribed in triangle $ABC$. Let $P$ be the point of contact of the circle and the side $\overline{AB}$ and $Q$ be the point on the circle, diametrical to the point $P$. We draw the line $CQ$ which intersects the side $\overline{AB}$ at the point $R$. Prove that the midpoint of the length $\overline{AB}$ is also the midpoint of the length $\overline{PR}$.

[nnosipov]

rsoldo в сообщении #1662812 писал(а):

the midpoint of the length $\overline{AB}$

Perhaps, "the midpoint of the segment $AB$" would be better. Anyway, the problem is very simple if we use algebraic calculations. I (weakly) hope a natural geometric solution will be more intriguing.

[wrest]

P - точка касания вписанной окружности.
R - точка касания вневписанной окружности - вот это и надо показать, видимо. Отсюда последует то что надо доказать в задаче.