In fig. XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that, XA + AR = XB + BR.
Since lengths of tangents from an exterior point to a circle are equal.
∴ XP = XQ …. (i) [From X]
AP = AR …. (ii) [From A]
BQ = BR …. (iii) [From B]
Now, XP = XQ
⇒ XA + AP = XB + BQ
⇒ XA + AR = XB + BR [Using equations (i) and (ii)]
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