Question
Mon June 18, 2012 By: Sonali

Prove that the transverse common tangents drawn to two congruent circles bisects the line segment joining their centres.

Expert Reply
Tue June 19, 2012
Let CD be the transverse common tangent to the two congruent circles with centres A and B.
Let AB and CD intersects at point E.
 
We know that the tangent at a point to a circle is perpendicular to the radius through the point of contact.
So, angle ACE = angle BDE = 90o
In triangles ACE and BDE, we have:
angle ACE = angle BDE = 90o
angle AEC = angle BED     (vertically opposite angles)
 
So, the triangles ACE and BDE are similar.
Therefore,
AC/BD = AE/BE
 
But the two triangles are congruent, their radii will be equal. So, AC = BD
Hence, AE = BE
 
Thus, the required result.
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