Ireland_2017

First, draw the circle centre $P$ through $A$ and let $D$ be the second intersection point of this circle with the line $AP$. Then construct the two lines that pass through $D$ and are parallel to the legs of the given angle. They intersect the legs of the angle at $B$ and $C$.



By construction, $P$ is the mid-point of $AD$ and $ABDC$ is a parallelogram. Because the diagonals of a parallelogram intersect each other at their mid-points, we obtain $|PB|=|PC|$.