uestion,please help

Question

In a triangle ABC ,D is a point on the side AB such that CD²=AD×DB,if sinA×sinB=Sin²C/2 then prove that Cd is internal bisector of angle C?

Answer 1

Hints for the proof

1. CD²=AD×DB ⇔ m∠C = 90°, CD ⊥ AB (Geometric Mean Theorem)

2. sinA×sinB=Sin²C/2 ⇒ m∠A = m∠B = 45° ⇒ m∠ACD = m∠BCD = 45° ⇒ CD is a bisector of ∠C.