A, B are the points on ⨀(O, r) such that tangents at A and B to the circle intersect in P. Show that the circle with as a diameter passes through A and B.

Given that A and B are points on circle (O, r) such that tangents at A and B to the circle intersect in P.

We have to prove that the circle with OP as a diameter passes through A and B.

Proof:

We know that a tangent drawn to a circle is perpendicular to the radius drawn from the point of contact.

⇒ OA ⊥ AP and OB ⊥ PB

⇒ ∠OAP = 90° and ∠OBP = 90°

∴ ∠OAP + ∠OBP = 180° … (1)

For OAPB,

∠OAP + ∠APB + ∠AOB + ∠OBP = 360°

⇒ (∠APB + ∠AOB) + (∠OAP + ∠OBP) = 360°

From (1),

⇒ ∠APB + ∠AOB + 180° = 360°

∴ ∠APB + ∠AOB = 180° … (2)

From (1) and (2),

OAPB is cyclic.

Here, OP makes a right angle at A.

Then OP is a diameter.

∴ The circle with OP as a diameter passes through A and B.

Hence proved.