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Two circles touch externally at a point $ P $. From a point $ T $ on the tangent at $ P $, tangents $ T O $ and TR are drawn to the circles with points of contact $ Q $ and $ R $ respectively. Prove that $ TQ = TR $."


Given:

Two circles touch externally at a point \( P \). From a point \( T \) on the tangent at \( P \), tangents \( T O \) and TR are drawn to the circles with points of contact \( Q \) and \( R \) respectively.

To do:

We have to prove that \( TQ = TR \).

Solution:


From the point $T, TR$ and $TP$ are two tangents to the circle with centre $O$.

This implies,

$TR = TP$....….(i)

Similarly,

From the point $T, TQ$ and $TP$ are two tangents to the circle with centre $C$.

$TQ = TP$...….(ii)

From (i) and (ii), we get,

$TQ = TR$

Hence proved.

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Updated on: 10-Oct-2022

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