Given: From a point P. Outside the circle with centre O, PA and PB are tangents drawn and ∠APB = 120°
And, OP is joined.
Required to prove: OP = 2 AP
Construction: Take mid-point M of OP and join AM, join also OA and OB.
Proof:
In right ∆OAP,
∠OPA = 1/2∠APB = 1/2 (120°) = 60°
∠AOP = 90° – 60° = 30° [Angle sum property]
M is mid-point of hypotenuse OP of ∆OAP [from construction]
So, MO = MA = MP
∠OAM = ∠AOM = 30° and ∠PAM = 90° – 30° = 60°
Thus, ∆AMP is an equilateral triangle
MA = MP = AP
But, M is mid-point of OP
So,
OP = 2 MP = 2 AP
– Hence proved.