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 **PT**^{2}** = 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.

∴ AM^{2} = AX × AY ...(1)

AN is the tangent and AXY is a secant.

∴ AN^{2} = AX × AY ...(2)

From (1) and (2), we have

AM^{2 }= AN^{2}

∴ AM = AN

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