Ceva and Menelaus Theorems for Angle Bisectors
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A Mathematical Droodle


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Copyright © 1996-2017 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

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

Menelaus and Ceva

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