Fontené Theorems/First

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 $A_2, B_2, C_2$.

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Proof

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Also see

• Second 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