perplexus.info :: Geometry : Concyclic Points of Tangency

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.

Submitted by Bractals

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Solution: (Hide)

Let the centers of the four circles be A, B, C,
and D. The quadrilateral ABCD has an incircle
since |AB|+|CD| = |BC|+|DA|.
The tangency points of the circles are the
points of tangency of the incircle with the
sides of ABCD.

QED

Note 1: See Jer's post for an alternate solution.
             See my reply for modifications.

Note 2: For another method - invert the configuration
             with one of the points of tangency as the center
             of inversion. Then all that is needed is to show
             that the images of the other three points of
             tangency are collinear.
             Which is quite easy.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: Solution	Bractals	2013-07-04 14:29:19
Solution	Jer	2013-07-04 01:27:18

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