Question

if PA and PB are tangents segments drawn from and external point P to a circle with center O prove that angel OAB=1/2of angle APB​

Answer 1

Case 1: Join PO and produce PO to Q. Then OA = OB = OP (radii), so we have two isosceles triangles OAP and OAQ.

LetOAP= α and OBP = β.ThenAPO= α(base angles of isosceles OAP)andBPO= β(base angles of isosceles OBP).HenceAOQ= 2α(exterior angle of OAP)andBOQ= 2β(exterior angle of OBP),AOB= 2α + 2β = 2(α + β) = 2 × APB.