Question

In $\Delta$ ABC, angle ABC is equal to twice the angle ACB, and bisector of angle ABC meets the opposite side at point P. Show that:
(i) CB: BA =CP: PA
(ii)AB $\times$ BC = BP $\times$ CA

Answer 1

1) In ΔABC, ∠ABC = 2∠ACB
Let ∠ACB = x
⇒∠ABC = 2∠ACB = 2x
Given BP is bisector of ∠ABC
Hence ∠ABP = ∠PBC = x
Using the angle bisector theorem, that is,
the bisector of an angle divides the side opposite to it in the ratio of other two sides.
Hence, CB : BA= CP:PA.

2) Consider ΔABC and ΔAPB

∠ABC = ∠APB [Exterior angle property]

∠BCP = ∠ABP [Given]

∴ ΔABC ≈ ΔAPB [AA criterion]

∴[Corresponding sides of similar triangles are proportional.]

⇒ AB x BC = BP x CA