Question 109
Let $ABC$ be an acute-angled triangle. Point $Z$ on the altitude of vertex $A$ and points $X$ and $Y$ on the extensions of the altitudes of vertices $B$ and $C$ are selected such that,
$$
\angle AYB = \angle BZC = \angle CXA = 90^\circ.
$$
Prove that $X$, $Y$ and $Z$ are collinear if and only if the length of the tangent from vertex $A$ to the nine-point circle of the triangle is equal to the sum of the lengths of tangents from vertices $B$ and $C$ to this circle.
$$
\angle AYB = \angle BZC = \angle CXA = 90^\circ.
$$
Prove that $X$, $Y$ and $Z$ are collinear if and only if the length of the tangent from vertex $A$ to the nine-point circle of the triangle is equal to the sum of the lengths of tangents from vertices $B$ and $C$ to this circle.