Fontené Theorems

Theorem

Let $\triangle ABC$ be a triangle.

Let $P$ be an arbitrary point in the same plane as $\triangle ABC$.

Let $A_1$, $B_1$ and $C_1$ be the midpoints of $BC$, $CA$ and $AB$ respectively.

Let $A_2 B_2 C_2$ be the pedal triangle of $P$ with respect to $\triangle A B C$.

Let $X, Y, Z$ be the intersections of $B_1 C_1$ and $B_2 C_2$, $A_1 C_1$ and $A_2 C_2$, and $A_1 B_1$ and $A_2 B_2$ respectively.

Then $A_2 X$, $B_2 Y$ and $C_2 Z$ concur at the intersection of the circle through $A_1, B_1, C_1$ and the circle through $A_2, B_2, C_2$.

Let $\triangle ABC$ be a triangle.

Let $P$ be a point moving on a fixed straight line through the circumcenter $O$ of $\triangle ABC$.

Then the pedal circle of $P$ with respect to passes through a fixed point $F$ on the Feuerbach circle of $\triangle ABC$.

Let $\triangle ABC$ be a triangle.

Let $P$ be an arbitrary point in the plane of $\triangle ABC$.

Let the isogonal conjugate of $P$ with respect to to $\triangle ABC$ be denoted $P'$.

Let $O$ be the circumcenter of $\triangle ABC$.

Then the pedal circle of $P$ is tangent to the nine point circle of $\triangle ABC$ if and only if $O$, $P$, $P'$ are collinear.

This entry was named for Georges Fontené.

Weisstein, Eric W. "Fontené Theorems." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FonteneTheorems.html