Question

If AC is the bisector of angle A and angle C and angle 1 equal to angle 2 , show that angle A equal to angle C

Answer 1

In a quadrilateral ABCD, if the angle bisectors of angle A and angle C meet on BD, how can I prove that the angle bisectors of angle B and angle D meet on AC?

1 ANSWER



Makarand Apte (मकरंद आपटे), Works on Computational Geometry

Answered Sep 25 2017 · Author has 607 answers and 1.7m answer views

The key to prove this is to use the angle bisector theorem, which states that the internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle bisected.



Since the bisectors of ∠A∠A and ∠C∠C meet at a common point (say XX) on the diagonal BDBD, it means that the diagonal BDBD is divided in a ratio BX:XCBX:XC which is equal to both AB:ADAB:AD and BC:CDBC:CD. So we have,
ABAD=BCCDABAD=BCCD
Rearranging, we have