"Prove that the external bisectors of the angles of a triangle meet the opposite sides in three collinear points. I need to prove this using only Menelaus Theorem, Stewart's Theorem, Ceva's Theorem. What I did:I tried by making a simple case diagram that is a diagram with obtuse angle in the given triangle. Then using Menelaus on angle bisectors with respect to the triangles and using angle bisector theorem for ratios of values."

raapjeqp 2022-10-15 Answered
Prove that the external bisectors of the angles of a triangle meet the opposite sides in three collinear points.
I need to prove this using only Menelaus Theorem, Stewart's Theorem, Ceva's Theorem.
What I did:I tried by making a simple case diagram that is a diagram with obtuse angle in the given triangle. Then using Menelaus on angle bisectors with respect to the triangles and using angle bisector theorem for ratios of values.
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Answers (1)

bargeolonakc
Answered 2022-10-16 Author has 16 answers
Let the triangle be A B C and external angle bisector of A B C cut A C in X, of A C B cut A B in Y, of B A C cut B C in Z.
By angle bisector theorem,
A X X C = A B B C . . . ( 1 )
C Z Z B = C A A B . . . ( 2 )
B Y Y A = B C C A . . . ( 3 )
(1) ×(2) ×(3) gives,
A X . C Z . B Y X C . Z B . Y A = 1
Therefore by converse of Menelaus Theorem X, Y, Z are collinear.
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