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Fontené Theorems

There are three theorems related to pedal circles that go under the collective title of the Fontené theorems.

The first Fontené theorem lets be a triangle and an arbitrary point, be the medial triangle of , and be the pedal triangle of with respect to . Denote the intersections of the corresponding sides of and as , , and (e.g., is the intersection of and , etc.), then the lines , and meet at a point common to the circumcircles of (which is the nine-point circle of ) and (which is the pedal circle of with respect to ).

The second Fontené theorem states that 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, as illustrated above.

The third Fontené theorem states that the pedal circle of a point is tangent to the nine-point circle iff and its isogonal conjugate lie on a line through the circumcenter. (Note that Johnson (1929) erroneously states this theorem with the orthocenter in place of the circumcenter.) Feuerbach's theorem is a special case of this theorem.

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

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References

Bricard, R. "Note au sujet de l'article précédent." Nouv. Ann. Math. 6, 59-61, 1906.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 52, 1971.Fontené, G. "Extension du théorème de Feuerbach." Nouv. Ann. Math. 5, 504-506, 1905.Fontené, G. "Sur les points de contact du cercle des neuf point d'un triangle avec les cercles tangents aux trois côtés." Nouv. Ann. Math. 5, 529-538, 1905.Fontené, G. "Sur le cercle pédal." Nouv. Ann. Math. 65, 55-58, 1906.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.

Fontené Theorems

Cite this as:

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