In Fig. 6.56, PS is the bisector of ∠QPR of Δ PQR. Prove that

 Construct a line segment RT parallel to SP which intersects the extended line segment QP at point T

Given that, PS is the angle bisector of ∠QPR.

∠QPS = ∠SPR     (i)

By construction,

∠SPR = ∠PRT (As PS || TR)     (ii)

∠QPS = ∠QTR (As PS || TR)    (iii)

Using these equations, we get:

∠PRT = ∠QTR

PT = PR

By construction,

PS || TR

By using basic proportionality theorem for ΔQTR,