Four disks are arranged in a plane such that each is externally tangent to two others. Prove that the four points of tangency lie on a circle.

call the centers of the disks A, B, C, D
create quadrilateral ABCD
call the four angle measures 2a, 2b, 2c, 2d

2a+2b+2c+2d=360 (angles of a quad. add to 360)
a+b+c+d=180

Connect the four points of tangency.  This forms four isoceles triangles with verticies A, B, C, D and also a new quadrilateral.  The base angles of the isoceles triangles are 90-a, 90-b, 90-c, 90-d respectively.

The new quadrilateral, therefore has angles a+b, b+c, c+d, d+a.

Both pairs of opposite angles of this quadrilateral sum to a+b+c+d=180

This is a sufficient condition for a quadrilateral to be incribed in a circle, therefore the points of tangency lie on a circle.

-Jer

(Do I need to explain the sufficient condition?)

Posted by Jer on 2005-02-02 17:32:27