# 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_{a}P_{b}P_{c} known as the *cevian triangle* of P with respect to the triangle ABC. By construction, triangles ABC and P_{a}P_{b}P_{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*.

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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

- 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
- Desargues' Hexagon
- 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 Cradle
- Cevian Cradle II
- Cevian Nest
- Cevian Triangle
- An Application of Ceva's Theorem
- Trigonometric Form of Ceva's Theorem
- Two Proofs of Menelaus Theorem
- Simultaneous Generalization of the Theorems of Ceva and Menelaus
- Menelaus from 3D
- Terquem's Theorem
- Cross Points in a Polygon
- Two Cevians and Proportions in a Triangle, II
- Concurrence Not from School Geometry
- Two Triangles Inscribed in a Conic - with Elementary Solution
- From One Collinearity to Another
- Concurrence in Right Triangle
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