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FROM A POINT P,TWO TANGENTS PA AND PB ARE DRAWN TO A CIRCLE O.IF THE DISTANCE BETWEEN P AND O IS EQUAL TO THE DIAMETER OF THE CIRCLE,SHOW THAT TRIANGLE APB IS EQUILATERAL.

Given that OP = 2OA (OP is equal to the length of the diameter, OA is the radius)

OP = Diameter

Also that OP is the hypotenuse of the right triangle OAP and Q is the midpoint of OP.

Since the midpoint of hypotenuse of a right triangle is equidistant from the vertices, we have

OA = AQ = OQ

Thus,

Now consider the triangle PAB.

Since PA = PB, we have,

Since , we have,

Hence triangle APB is an equilateral triangle.