Two unit circles are externally tangent at B. Let line AB and lineBC be diameters of the two circles. A tangent is drawn from A to the circle with diameter line BC, and a tangent is drawn from C to the circle with diameter AB so that the two tangent lines are parallel. Find the distance between the two lines of tangency
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Let M be the point of tangency on the AB circle, draw the line between A and C and the part diameter from M through the centre O of the AB circle, meeting the 'A' tangent at N.
There's a well known theorem concerning the tangents to a circle from an outside point, in this instance we can say that AC.BC = CM.CM.
That gets you the length of CM and so you know everything about the triangle CMO.
The triangle ANO is similar to the triangle CMO so we can deduce the length of ON, etc..