Question

In a triangle ABC, the incircle (centre O) touches BC, CA and AB at points P, Q and R respectively. Calculate:

i)${\displaystyle \angle }$QPR .

Answer 1

The incircle touches the sides of the triangle ABC and
OP ⊥ BC,OQ ⊥ AC,OR ⊥ AB

i) Now arc RQ subtends${\displaystyle \angle }$QOR at the centre and ${\displaystyle \angle }$QPR at the remaining part of the circle. 

∴ ${\displaystyle \angle }$QPR = ${\displaystyle \frac{1}{2}}$ ${\displaystyle \angle }$ QOR

⇒ ${\displaystyle \angle }$QPR = ${\displaystyle \frac{1}{2}\times 120°}$

⇒ ${\displaystyle \angle }$ QPR = 60°