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Two circles with radius 1 are externally tangent at B, and have line AB and line BC as diameters. A tangent to the circle with diameter BC passes through A, and a tangent to the circle with diameter AB passes through C, so that the tangent lines are parallel. Find the distance between the two tangent lines. 

 Nov 23, 2020
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Let the center of circle with diameter AB be O. Let the circle with diameter BC have center Q.

 

Let the tangent to circle O be M. Let the tangent to circle Q be N.

 

Then $\triangle QNA$ is right. The length QA is 3, and it is the hypotenuse, and QN is a radius, so it equals 1.

 

If we extend QN downward until it intersects MC at point X, we have that our desired distance is XN.

 

By AA similarity, $\triangle ANQ \sim CXQ,$ so $\frac{NQ}{AQ} = \frac{QX}{QC} \Longrightarrow \frac{1}{3} = QX \Longrightarrow QX + DQ = \frac{4}{3} !!!!$

 Nov 23, 2020
edited by Pangolin14  Nov 23, 2020

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