Miquel Point

Unlike the circumcenter or incenter, the Miquel point generally depends on the chosen points $D$, $E$, and $F$, not just on triangle $ABC$.

Consider four lines in general position, with no two parallel and no three concurrent.

Each choice of three lines determines a triangle, giving four triangles in total.

The circumcircles of these four triangles pass through a common point.

This is known as the Four-Line Miquel Theorem.

Let $ABC$ be an acute triangle. Let $D$, $E$, and $F$ be the feet of the altitudes from $A$, $B$, and $C$, respectively, and let $H$ be the orthocenter. Prove that the Miquel point of $AEF$, $BFD$, and $CDE$ is $H$.

Let the incircle of triangle $ABC$ touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Let $I$ be the incenter. Prove that the Miquel point of $AEF$, $BFD$, and $CDE$ is $I$.