First slide

Analysis of two circles in circles

Question

Three circles with radii 3cm, 4cm and 5cm touch each other externally. If A is the point of intersection of tangents to these circles at their points of contact, then the distance of A from the points of contact is

Very difficult

A

$\sqrt{3}$
B

$2$
C

$\sqrt{5}$
D

$\sqrt{6}$

Solution

It is clear that A is the centre of the incircle of the triangle PQR formed by the lines joining the centres of the given circles. We have $P Q = 5 + 4 = 9,$ $Q R = 4 + 3 = 7,$ $R P = 3 + 5 = 8,$ distance of A from the points of contact is the radius r of the incircle of triangle PQR.
From trigonometry $r = Δ / s$ where $r = Δ / s$ is the area and $2 s$ is the perimeter of the triangle PQR
$Δ = \sqrt{12 (12 - 7) (12 - 8) (12 - 9)} = \sqrt{12 \times 5 \times 4 \times 3}$
$= 12 \sqrt{5}$
$s = (1 / 2) (9 + 7 + 8) = 12,$
Hence $r = \sqrt{5}$

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