Fontené Theorems

1. If the sides of the Pedal Triangle of a point $P$ meet the corresponding sides of a Triangle $\Delta O_1O_2O_3$ at $X_1$, $X_2$, and $X_3$, respectively, then $P_1X_1$, $P_2X_2$, $P_3X_3$ meet at a point $L$ common to the Circles $O_1O_2O_3$ and $P_1P_2P_3$. In other words, $L$ is one of the intersections of the Nine-Point Circle of $A_1A_2A_3$ and the Pedal Circle of $P$.

2. If a point moves on a fixed line through the Circumcenter, then its Pedal Circle passes through a fixed point on the Nine-Point Circle.

3. The Pedal Circle of a point is tangent to the Nine-Point Circle Iff the point and its Isogonal Conjugate lie on a Line through the Orthocenter. Feuerbach's Theorem is a special case of this theorem.

See also Circumcenter, Feuerbach's Theorem, Isogonal Conjugate, Nine-Point Circle, Orthocenter, Pedal Circle

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 245-247, 1929.