Question

if from any point on the chord of two interesting circles, tangents be drawn to the circles. prove that they are equal.

Answer 1

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) [ From the theorem]
AN is the tangent and AXY is a secant.
∴ AN2 = AX × AY ..................(2) [ From the theorem]
From (1) and (2), we get
AM2 = AN2
∴ AM = AN