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the main idea is to find the distance b/w the foot of perpendiculars from the center of the two given circles to the common tangent by two ways and then equating the two equations obtained to get the value of radius.
1st equation
form a right triangle by taking the distance b/w the centers of the circles as hipotenuse and solve for the base using pythagores th.
2nd equation
form right angled triangles (2) by joining the centre of the required circle with the centres of the two given circles and solve for the base and then sum up the two distances and then equate it with the distance obtainesd in the first equation
r = r1 * r2/( sq. root r1+ sq. root r2)2
1/sq root r = (sq. root r1+ sq. root r2)/ (sq root (r1*r2))
we see that
1/sq root r = 1/sq root r1 + 1/sq root r2
hope it helps
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