if from any point on the common chord of two intersecting circles , tangents be drawn to the circles , prove that they are equal .
Let PT be a tangent to the circle from an external point P and a secant to the circle through P intersects the circle at points A and B, then PT2 = PA × PB
This property is used to solve the given question.
Let the two circles intersect at points X and Y. XY is the common chord.
Suppose A is a point on the common chord and AM and AN be the tangents drawn from A to the circle.
AM is the tangent and AXY is a secant.
∴ AM2 = AX × AY ...(1)
AN is the tangent and AXY is a secant.
∴ AN2 = AX × AY ...(2)
From (1) and (2), we have
AM2 = AN2
∴ AM = AN
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