Collinearity

In any non-isosceles triangle ABC, the bisectors of the exterior angles at A, B, and C meet the opposite sides at points D, E, and F respectively. Prove that D, E, and F are collinear."

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## Friday, December 5, 2008

### Triangle with the bisectors of the exterior angles.

Labels:
angle bisector,
collinear,
Menelaus' theorem,
triangle

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AD is the external bisector of angle A. Hence, BD/DC = - AB/AC. Likewise, CE/AE = -BC/AB & AF/BF=-AC/BC. Now (AF/BF)*(BD/DC)*(CE/AE)=(-AC/BC)*(-AB/AC)*(-BC/AB)= -1. Therefore, by the converse of Menelaus's Theorem D, E & F are collinear. Would that be correct, Antonio?

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