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Given: A circle with centre O; PA and PB are two tangents to the circle drawn from an external point P.
To prove: PA = PB
Construction: Join OA, OB, and OP.

It is known that a tangent at any point of a circle is perpendicular to the radius through the point of contact.
OA
PA and OB
PB ... (1)
In
OPA and
OPB:
OAP =
OBP (Using (1))
OA = OB (Radii of the same circle)
OP = OP (Common side)
Therefore,
OPA
OPB (RHS congruency criterion)
PA = PB
(Corresponding parts of congruent triangles are equal)
Thus, it is proved that the lengths of the two tangents drawn from an external point to a circle are equal.
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