Ceva and Menelaus Theorems for Angle Bisectors
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A Mathematical Droodle
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Copyright © 1996-2018 Alexander Bogomolny
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At each vertex of a triangle there is a couple of angle bisectors: a bisector of the interior angle and a bisector of the exterior angle. It's well known that the bisectors taken one for each vertex are concurrent provided none or two of the bisectors are external. This is a particular case of Ceva's theorem. In general, every angle bisector crosses the opposite side of the triangle (or its extension). It then follows from the Menelaus theorem, that every three such points are collinear provided none or two of the bisectors are internal.
References
- R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.
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
|Activities| |Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny
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