Problem: In the adjoining figure, every point of circle is exterior to circle . Let and be the points of intersection of an internal common tangent with the two external common tangents. Then the length of is
Answer Choices:
A. The average of the lengths of the internal and external common tangents
B. Equal to the length of an external common tangent if and only if circles and have equal radii
C. Always equal to the length of an external common tangent
D. Greater than the length of an external common tangent
E. The geometric mean of the lengths of the internal and external common tangents
Solution:
In the adjoining figure, and are the points of tangency of the external common tangents; and and are the points of tangency of the internal common tangent.
From the fact that the tangents to a circle from an external point are equal, we obtain:
So , and thus .
Since .
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