OI line property

[ins-]

In the triangle $ABC$ is drawn a line $l$ through its incenter and its circumcenter. Through tangency points of the incircle with the sides - $D$, $E$, $F$ are drawn three parallel lines that intersects $l$ at the points $D'$, $E'$, $F'$. Prove that one of the segments $DD'$, $EE'$, $FF'$ is equal to the sum of the other two.

[ins-]

In the triangle $ABC$ with circumcenter $O$ and ncenter $I$ - $D$, $E$, $F$ are the tangency points of its incircle with its sides. $H$ and $G$ are the orthocenter and the center of gravity of the triangle $DEF$. Prove that $O$, $I$, $G$, $H$ are collinear.