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Radical Axis

Three circles...

Question

Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from a point of contact is $4$ . Then the ratio of the area of triangle formed by centers to the sum of the radii of the circles is

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Solution

Let $C_{1}, C_{2}$ and $C_{3}$ be the centres of three circles of radii $r_{1}, r_{2}$ and $r_{3}$ respectively, then the length, then

From above figure in $△ C_{1} C_{2} C_{3}$
$C_{1} C_{2} = r_{1} + r_{2}, C_{2} C_{3} = r_{2} + r_{3}, C_{1} C_{3} = r_{1} + r_{3}$

Now, area of the trianlge $△ C_{1} C_{2} C_{3}$
$= Area of (△ O C_{2} C_{3} + △ O C_{1} C_{3} + △ O C_{1} C_{2})$
as $O P = O Q = O R = 4$
$∴ Area of △ C_{1} C_{2} C_{3} = 4 (r_{1} + r_{2} + r_{3})$

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