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 DeltaABC be a triangle and P an arbitrary point, DeltaA^'B^'C^' be the medial triangle of DeltaABC, and DeltaXYZ be the pedal triangle of DeltaABC with respect to P. Denote the intersections of the corresponding sides of DeltaA^'B^'C^' and DeltaXYZ as D, E, and F (e.g., D is the intersection of B^'C^' and YZ, etc.), then the lines XD, YE and ZF meet at a point L common to the circumcircles of DeltaA^'B^'C^' (which is the nine-point circle of DeltaABC) and DeltaXYZ (which is the pedal circle of DeltaABC with respect to P).

Fontenes second theorem

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 P is tangent to the nine-point circle iff P and its isogonal conjugate P^' 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.

See also

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

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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.

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Fontené Theorems

Cite this as:

Weisstein, Eric W. "Fontené Theorems." From MathWorld--A Wolfram Web Resource.

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