Question

prove that the angle bisector of triangle divides the opposite side in the ratio of sides containing the angle.

Answer 1

The internal (external) bisector of an angle of a triangle divides the opposite side internally (externally) in the ratio of the corresponding sides containing the angle.



Let E be the intersection of AD and the line parallel to AB through C. ∠AEC = ∠BAE (Transversal theorem: the line that cuts two parallels, cuts it under equal angles), meaning that ΔACE is isosceles and thus AC = CE. In addition, triangles ABD and ECD are similar, implying

AB : CE = BD : CD.

Together with CE = AC, we obtain the required AB : AC = BD : CD.



Proof (Externally) :

Answer 2

Answer:

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