Cevian Triangle

For a given point P in the plane of a triangle ABC, the feet of the cevians through P form a triangle P_aP_bP_c known as the cevian triangle of P with respect to the triangle ABC. By construction, triangles ABC and P_aP_bP_c 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

• Desargues' Theorem
• 2N-Wing Butterfly Problem
• Cevian Triangle
• Do You Speak Mathematics?
• Desargues in the Bride's Chair (with Pythagoras)
• Menelaus From Ceva
• Monge from Desargues
• Monge via Desargues
• Nobbs' Points, Gergonne Line
• Soddy Circles and David Eppstein's Centers
• Pascal Lines: Steiner and Kirkman Theorems II
• Pole and Polar with Respect to a Triangle
• The Lepidoptera of the Circles

Menelaus and Ceva

• The Menelaus Theorem
• Menelaus Theorem: proofs ugly and elegant - A. Einstein's view
• Ceva's Theorem
• Ceva in Circumscribed Quadrilateral
• Ceva's Theorem: A Matter of Appreciation
• Ceva and Menelaus Meet on the Roads
• Menelaus From Ceva
• Menelaus and Ceva Theorems
• Ceva and Menelaus Theorems for Angle Bisectors
• Ceva's Theorem: Proof Without Words
• Cevian Nest
• Cevian Triangle
• An Application of Ceva's Theorem
• Trigonometric Form of Ceva's Theorem
• Two Proofs of Menelaus Theorem
  

  

  Copyright © 1996-2009 Alexander Bogomolny