About an angle bisector

Question

In any triangle ABC, the bisector of angle ABC intersects AC at D. Prove that the product of the lengths of AB and BC is equal to the square of the length of BD plus the product of the lengths of AD and CD.

Answer 1

This answer is for Mark M.

I finally got it!

I didn't need the angle bisector theorem.

But I did need to draw the circumscribing circle and extend BD to E on the circle.

Then I used the theorem about intersecting chords and also similar triangles ABD and BEC along with some simple algebra.

Thanks for your interest.

Answer 2

By the Angle Bisector Theorem

AD / DC = AB / BC