SE-SKOLORNAS 2009 Problem 5

A semicircular arc and a diameter $AB$ with a length of $2$ are given. Let $O$ be the midpoint of the diameter. On the radius perpendicular to the diameter, we select a point $P$ at the distance $d$ from the midpoint of the diameter $O$ , $0 <d <1$ . A line through $A$ and $P$ intersects the semicircle at point $C$ . Through point $P$ we draw another line at right angle against $AC$ that intersects the semicircle at point $D$ . Through point $C$ we draw a line $l_1$ , parallel to $PD$ and then a line $l_2$ , through $D$ parallel to $PC$ . The lines $l_1$ and $l_2$ intersect at point $E$ . Show that the distance between $O$ and $E$ is equal to $\sqrt{2- d^2}$