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 Concyclic Points of Tangency (Posted on 2013-07-02)

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 No Rating

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 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 180º 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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