In the given fig, PS is the bisector of ∠QPR of ∆PQR. Prove that 

In the given fig., D is a point on hypotenuse AC of ∆ABC, DIM ⊥ BC and DN ⊥ AB. 
Prove that:
(i) DM² = DN × MC.
(ii) DN² = DM × AN.

In the given fig., ABC is a triangle in which ∠ABC > 90° and AD ⊥ CB produced. Prove that AC² = AB² + BC² + 2 BC . BD.

234.

In the given Fig,  ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC. Prove that AC²= AB² + BC² - 2 BC.BD.

In the given fig, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:
(i)    
(ii) 
(iii)   

In the given fig, two chords AB and CD intersect each other at the point P. Prove that:
(i) ∆APC ~ ∆DPB.
(ii) AP . PB = CP . DP

In the given Fig, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that
(i) ∆PAC ~ ∆PDB.
(ii) PA. PB = PC . PD.

In the given fig, D is a point on side BC of ∆ABC such that   Prove that AD is the bisector of ∠BAC.