KöMaL 2019 B5028

Let us define a function $f$ as follows. For any acute-angled triangle $XYZ$, if $P$ is a point on $YZ$, then $f(P;XYZ)$ is defined as the line joining the feet of perpendiculars from $P$ to lines $XY$; $XZ$. Let $ABC$ be a triangle with orthocenter $H$. Let $A'B'C'$ be the orthic triangle of $ABC$. Let $A''\equiv f(B';HCA) \cap f(C';HAB)$. Similarly, points $B''$; $C''$ are defined. Show that the lines $AA''$; $BB''$; $CC''$ are concurrent.