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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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