Fontené Theorems/Third

Theorem

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 $P^{-1}$.

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

Then the pedal circle of $P$ is tangent to the Feuerbach circle of $\triangle ABC$ if and only if $O$, $P$, $P^{-1}$ are collinear.

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Proof

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By the Second Fontené Theorem we can prove that the second intersection $Q'$ of the circle $O'$ and the circle $E$ is the anti-Steiner point of $OP^{-1}$.

This means $Q' = Q$ if and only if $O P = O P^{-1}$

That is:

$O$, $P$ and $P^{-1}$ are collinear.

$\blacksquare$

Also see

• First Fontené Theorem
• Second Fontené Theorem

Source of Name

This entry was named for Georges Fontené.

Sources

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