Two circles with centres $M$ and $N$ intersect each other at $P$ and $Q$ . The tangents drawn from point $R$ on the line $P Q$ touch the circles at $S$ and $T$ . Prove that, $R S = R T$ .

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Solution

Ref. Image.
In the situation shown, AP is a tangent to circle and QR is
any chord meeting A, by properly
of circles, $A P^{2} = Q A . R A$
In the given situation for circle I , $R S^{2}$ = RQ .QP
and for circle II, $R T^{2}$ =RQ.QP
$\Rightarrow R T^{2} = R S^{2} \Rightarrow$ RT = RS

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Two circles with centres M and N intersect each other at P and Q. The tangents drawn from point R on the line PQ touch the circles at S and T. Prove that, RS=RT.

Ref. Image.In the situation shown, AP is a tangent to circle and QR isany chord meeting A, by properlyof circles, AP2=QA.RAIn the given situation for circle I ,RS2 = RQ .QPand for circle II,RT2=RQ.QP⇒RT2=RS2⇒ RT = RS

Two circles with centres $M$ and $N$ intersect each other at $P$ and $Q$ . The tangents drawn from point $R$ on the line $P Q$ touch the circles at $S$ and $T$ . Prove that, $R S = R T$ .

Ref. Image.
In the situation shown, AP is a tangent to circle and QR is
any chord meeting A, by properly
of circles, $A P^{2} = Q A . R A$
In the given situation for circle I , $R S^{2}$ = RQ .QP
and for circle II, $R T^{2}$ =RQ.QP
$\Rightarrow R T^{2} = R S^{2} \Rightarrow$ RT = RS