Question

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

Answer 1

Let $2$ circle intersect at points $A$ and $B$ such that $AB$ is the common chord. Extend the common chord to point $P$.Let $PR$ and $PQ$ be two tangents respectively on the two circles (as shown in the figure).

Since $PQ$ is that tangent and $PAB$ is the secant for one circle,

$\therefore P{Q}^2=PB\times PA$...(i)

Similarly, $PR$ is the tangent and $PAB$ is the secant for the other circle,

$\therefore P{R}^2=PB\times PA$...(ii)

From (i) and (ii),

$\Rightarrow P{Q}^2=P{R}^2$

$\therefore PQ=PR$

Hence, the tangents drawn from any point of the chord onto circles are equal.