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Concyclic Points of Tangency (Posted on 2013-07-02) Difficulty: 3 of 5

Consider four circles each of which is
externally tangent to two of the others.

Prove that the four points of tangency
are concyclic.

See The Solution Submitted by Bractals    
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Solution Solution | Comment 1 of 2
For four points to be concyclic, they must be the vertices of a cyclic quadrilateral. If two opposite angles of a quadrilateral sum to 180 it is cyclic.  This is what I will show below.

Call the centers of the circles in order A, B, C, D.  Call points of tangency W,X,Y,Z so that we can draw quadrilaterals ABCD and WXYZ.

Being a quadrilateral makes angles A+B+C+D=360
There are 4 isosceles triangles which allow us to find: 
Angle AWZ = 180-A/2
Angle BWC = 180-B/2 etc.
Angle ZWX = 180-(A+B)/2
Angle XYZ = 180-(C+D)/2 etc.

Angles ZWX and XYZ are the opposite angles mentioned in the first paragraph.  Their sum is
360-(A+B+C+D)/2 = 360 - 360/2 = 180
QED

  Posted by Jer on 2013-07-04 01:27:18
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