Question 59
Let $ABC$ be a triangle such that $|AB| \gt |AC|$. Let $P$ be the midpoint of the segment $\overline{BC}$ and $S$ the intersection of the bisector of the angle $\angle BAC$ and the segment $\overline{BC}$. The line parallel to the line $AS$, which passes through $P$, intersects the lines $AB$ and $AC$ in points $X$ and $Y$ respectively. Let $Z$ be a point such that $Y$ is the midpoint of the segment $XZ$, and that lines $BY$ and $CZ$ intersect at $D$.
Prove that the bisector of the angle $\angle BDC$ is parallel to the line $AS$.
(D. Monk, New Problems in Euclidean Geometry)
Prove that the bisector of the angle $\angle BDC$ is parallel to the line $AS$.
(D. Monk, New Problems in Euclidean Geometry)