ÖMO 2020 Problem 5

Let $h$ be a semicircle with diameter $AB$ . Let $P$ be an arbitrary point inside the diameter $AB$ . The perpendicular through $P$ on $AB$ intersects $h$ at point $C$ . The line $PC$ divides the semicircular area into two parts. A circle will be inscribed in each of them that touches $AB, PC$ and $h$ . The points of contact of the two circles with $AB$ are denoted by $D$ and $E$ , where $D$ lies between $A$ and $P$ . Prove that the size of the angle $DCE$ does not depend on the choice of $P$ .

(Walther Janous)