parmenides51 wrote: In a triangle $ABC$ the excircle at the side $BC$ touches $BC$ in point $D$ and the lines $AB$ and $AC$ in points $E$ and $F$ respectively. Let $P$ be the projection of $D$ on $EF$. Prove that the circumcircle $k$ of the triangle $ABC$ passes through $P$ if and only if $k$ passes through the midpoint $M$ of the segment $EF$. Trivially as $M$ lies on $\odot(ABC)$. Hence, $MB=MC$. Brianchon on $ABCDEF$ gives $AD,BE,CF$ are concurrent. Hence, id $EF\cap BC=T$. Then $(BC;DT)$ is harmonic. Hence, $PM$ is the Exterior angle bisector of $\angle BPC$. Hence, $P\in\odot(ABC)$. Similarly for the other direction. $\blacksquare$