Question
Sat February 23, 2013 By:

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

Expert Reply
Sat February 23, 2013
Let the two circles intersect at points M and M. MN is the common chord.

Suppose O is a point on the common chord and OP and OQ be the tangents drawn from A to the circle.

OP is the tangent and OMN is a secant.

According to the theorem which says that if PT is 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

Therefore, OP2 = OM*ON (1)
 
Similarly, OQ is the tangent and OMN is a secant for that also. 
Therefore OQ2 = OM*ON (2)
 
Comparing 1 and 2
 
OP2 = OQ2
i.e OP = OQ
Hence, the length of the 2 tangents is equal. Hence proved. 
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