Question

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

Answer 1

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 𝑃𝑇2 = 𝑃𝐴 × 𝑃𝐵"

Now AM is the tangent and AXY is a secant ∴ 𝐴𝑀2 = 𝐴𝑋 × 𝐴𝑌 … . . (𝑖)

AN is a tangent and AXY is a secant ∴ 𝐴𝑁2 = 𝐴𝑋 × 𝐴𝑌 … . . (𝑖𝑖)

From (i) & (ii), we have 𝐴𝑀2 = 𝐴𝑁2

∴ AM = AN