OP bisects  ∠AOC, OQ bisects ∠COB  and OP ⊥ OQ. Show that A, O, B are collinear.
    
  

Given: OP bisects ∠AOC
       OQ bisects ∠COB
       OP ⊥ OQ
To prove: A, O, B are collinear.
Proof: OP bisects ∠AOC
       ∠AOP =  ∠POC ⇒ ∠COP = 1/2 ∠COA --------------(1)
OQ bisects ∠BOC
       ∠COQ =  ∠QOB
⇒ ∠COQ = 1/2 ∠ BOC                ------------------(2)
    ∠POQ = ∠COP + ∠COQ =  90°  -------------------(3)
Substitute (1) and (2) in (3)
    1/2 ∠COA + 1/2 ∠ BOC =  90°
       1/2 [ ∠COA + ∠ BOC] = 90°
  ∠AOC + ∠COB  = 90° x 2 = 180°
=>  ∠AOC ,  ∠COB  form a linear pair.
.^..  A, O, B are collinear (Linear Pair Axiom).