Question 175
Let $\triangle ABC$ be an acute triangle such that $|AB| + |BC| = 4|AC|$ and $|AB| \lt |BC|$. Let $D$ be the intersection of the bisector of $\angle ABC$ with the side $AC$. Points $P$ and $Q$ lie on segment $BD$ such that $|BP| = 2|DQ|$. Let $\ell$ be the line through $P$ parallel to $AC$. The line through $Q$ perpendicular to $BD$ intersects the segments $AB$ and $BC$ in $X$ and $Y$, respectively. Suppose that both $X$ and $Y$ lie on the opposite side of $\ell$ to $B$.
An ant starts its journey in $X$ and goes from there to a point on $AC$, then a point on $\ell$, then back to a (possibly different) point on $AC$ and finally to $Y$. Prove that the length of the ant's shortest possible route is equal to $4|XY|$.
An ant starts its journey in $X$ and goes from there to a point on $AC$, then a point on $\ell$, then back to a (possibly different) point on $AC$ and finally to $Y$. Prove that the length of the ant's shortest possible route is equal to $4|XY|$.