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Subject: "Intersecting tangent circles"     Previous Topic | Next Topic
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Conferences The CTK Exchange College math Topic #639
Reading Topic #639
Bractals
Member since Jun-9-03
Jul-24-07, 08:00 PM (EST)
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"Intersecting tangent circles"
 
   Given the following four circles

(x-a)2 + y2 = a2

(x-b)2 + y2 = b2

x2 + (y-c)2 = c2

x2 + (y-d)2 = d2


where a,b,c, and d are nonzero real numbers with a≠b and c≠d.

Prove synthetically that the intersections of the four circles (that are not the origin) lie on a circle.

I have verified it using Geometer's Sketchpad.


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alexb
Charter Member
2058 posts
Jul-24-07, 10:03 PM (EST)
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1. "RE: Intersecting tangent circles"
In response to message #0
 
   This is a particular case of Clifford's Lemma (or, sometimes, Theorem), see

https://www.cut-the-knot.org/Curriculum/Geometry/CliffordTheorem.shtml


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Bractals
Member since Jun-9-03
Jul-25-07, 06:47 AM (EST)
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2. "RE: Intersecting tangent circles"
In response to message #0
 
   Thanks for the response Alex.

Just after I posted the problem, I realized that an inversion through a circle with center at the origin made the problem trivial.


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