Start by drawing a diagram including the line joining the centres of the two circles and radii of both circles meeting the common tangent(s) at 90 degrees ( you only really need to draw one of the tangents )
It is handy to construct a line segment parallel to the line of centres joining the point of tangency in the smaller circle to the radius of the bigger circle forming a parallelogram and a right angled triangle, That triangle is similar to a triangle whose vertices include the point of intersection that you want to find ( clearly this point lies on the line of centres, )
This similarity allows you to calculate the distance from the point of intersection to the point of contact between the two circles.
( Actually there are three points of intersection if you include the common tangent that passes through the point of contact - the count goes down to two if the circles have the same radius. )
Jan 18th, 2015
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