Problem
Let$\triangle DEF,~ \triangle IJK $ be the pedal triangle and the circumcevian triangle of $P$ with respect to $\triangle ABC$, $X,Y,Z$ be the points such that $D,E,F$ are the midpoints of $XI,YJ,ZK$, and $H$ be the orthocenter of $\triangle ABC.$ Show that $X,Y,Z,H$ are concyclic.
P.S
Indeed, the problem had been included in the book, Geometry in Figures by Arseniy Akopyan, which was numbered as 4.2.8). But it's a pity that there exists no proof all over the book.
I have tried so far but obtain nothing. Please give a geometric proof without much computation as far as possible. Thanks in advance.