Cevian Triangle

For a given point P in the plane of a triangle ABC, the feet of the cevians through P form a triangle PaPbPc known as the cevian triangle of P with respect to the triangle ABC. By construction, triangles ABC and PaPbPc are perspective from point P. By Desargues' theorem, they are also perspective from a line.

The orthic triangle is the cevian triangle of the orthocenter, and thus falls into the general framework. The line from which it is perspective to the base triangle ABC is known as the orthic axis.

The existence of the orthic axis could be proven in many ways. For example, R. Honsberger observes that, because of the mirror property of the orthic triangle, the side lines of triangle ABC form external bisectors of the orthic triangle. (I.e., they bisect the external angles of the latter.) It can be shown by Menelaus' theorem that, for any triangle, the points of intersection of the external bisectors with the opposite sides of the triangle are collinear.

References

  1. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, pp. 151

Desargues' Theorem

Menelaus and Ceva

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