Lotfi Zadeh Olympiad 2021 1p (Aze)

  geometry problems from Lotfi Zadeh Olympiad from Azerbaijan,  with aops links in the names

collected inside aops here

Lotfi Zadeh 2021

2021 Lotfi Zadeh p1

In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.