Help required to prove a question on triangles, collinearity and cyclic quadrilaterls

Question

In an acute $\triangle ABC$ $D,E,F$ are feet of perpendiculars from $A,B,C$ respectively. The perpendiculars from $F$ to $BC,AD,CA,BE$ intersect them at $X,Y,Z,V$.

How do I prove that $X,Y,Z,V$ are collinear points?

Answer 1

$F$ is a point on the circumcircle of $BEC$, hence $X,V,Z$ are collinear by Simson's theorem.

On the other hand, $F$ also belongs to the circumcircle of $ADC$, hence $X,Y,Z$ are collinear, too.