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If from any point on the common chord of two intersecting circles, tangents be drawn to circles, prove that they are equal.
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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 A to the circle
We need to show that AM = AN.
In order to prove the above relation, following property will be used.
“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 ЁЭСГЁЭСЗ2 = ЁЭСГЁЭР┤ × ЁЭСГЁЭР╡"
Now AM is the tangent and AXY is a secant ∴ ЁЭР┤ЁЭСА2 = ЁЭР┤ЁЭСЛ × ЁЭР┤ЁЭСМ … . . (ЁЭСЦ)
AN is a tangent and AXY is a secant ∴ ЁЭР┤ЁЭСБ2 = ЁЭР┤ЁЭСЛ × ЁЭР┤ЁЭСМ … . . (ЁЭСЦЁЭСЦ)
From (i) & (ii), we have ЁЭР┤ЁЭСА2 = ЁЭР┤ЁЭСБ2
∴ AM = AN
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