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Geometric Two Column Proof?
Given a triangle in which the bisector of one angle also bisects the opposite side, prove that the triangle is isosceles.
Let's label the sides to make it easier to talk about. In triangle ABC the bisector of angle A also bisects the opposite side BC at point D.
2 Answers
- steppenwolfLv 51 decade agoFavorite Answer
This follows directly from the angle bisector theorem. Refer to wikipedia for a proof.
- Scythian1950Lv 71 decade ago
Flipping ABC about the bisector creates mirror image A'BC', so that imposing A'BC' onto ABC "creates new isoceles triangles" APC' = A'PC, where P is the point of bisection of line AC. A little fiddling around with the angles quickly leads to a contradiction. There are several ways of showing a contradiction. One way is to show that the sums of the angles APB + BPA' + A'PC = 180 - 2ABP, which is impossible unless ABP = 0.
