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

$G i v e n t w o c i r c l e s w i t h c e n t r e s M a n d N i n t e r s e c t e a c h o t h e r a t P a n d Q . T h e t a n g e n t s d r a w n f r o m p o i n t R o n t h e l i n e P Q t o u c h t h e c i r c l e s a t S a n d T . w e h a v e t o p r o v e t h a t R S = R T B y t a n g e n t s e c a n t t h e o r e m w h i c h s t a t e s t h a t w h e n t a n g e n t a n d a s e c a n t c o n s t r u c t f r o m o n e s i n g l e e x t e r n a l p o i n t t o a c i r c l e t h e n s q u a r e o f l e n g t h o f t a n g e n t m u s t b e e q u a l t o t h e p r o d u c t o f l e n g t h s o f w h o l e s e c a n t s e g m e n t a n d t h e e x t e r i o r p o r t i o n o f s e c a n t s e g m e n t .$
In a circle $m$
tangent $r s = r q$ (as they originate from same point...........1)
In circle $n$
$r q = r t . . . . . . . . . . (2)$
form $(1)$ and $(2)$
$r s = r t$

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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.

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, $R S = R T .$