Fontené Theorems/Second

Theorem

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 $\triangle ABC$.

Proof

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By the First Fontené Theorem, the point of contact $Q$ of the circle $E$ and the circle $O'$ is the reflection of a point $F$ which lies on $O P$ with respect to the line $B_1 C_1$.

It is easy to show that $O$ is the orthocenter of triangle $A_1 B_1 C_1$

thus $Q$ is the anti-Steiner point of $d$.

Therefore $Q$ is fixed.

Also known as

The second Fontené theorem is also known as Griffiths' theorem, for John Griffiths.

Also see

• First Fontené Theorem
• Third 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