Puzzle: circle geometry – stumped!
June 11, 2007 am30 9:26 am
A friend offers me what he warrants is a famous puzzle:
given three circles, how would you construct a circle tangent to all three?
I’ve been going around in, well you know, not getting very far. I have managed to pose 2 additional questions, which I thought would help, but not yet!
- describe the circumstances in which this would be impossible
- instead of constructing the tangent, throw the whole problem onto the coordinate plane, and determine the equation of the circle tangent to the other three.
Geometry is a weak point for me, and constructions especially so.
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JD2718 
1. In projective geometry, you can always construct two extended circles (= a circle or a line) tangent to three given extended circles. So in normal geometry you can construct a tangent circle if and only if you can’t construct two lines tangent to all three circles, which you can check since every pair of circles has at most four mutual tangents.
2. Let the three circles be centered at (x(i), y(i)) and have radii r(i), i = 1, 2, 3. A circle centered at (x4, y4) with radius r4 is tangent to all three iff the distance between (x4, y4) and (x(i), y(i)) is |r4 +/- r(i)|, i
Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.
It is a famous problem: http://en.wikipedia.org/wiki/Descartes'_theorem
My dad and I walked through the proof by poem in late middle school and I thought it was the coolest thing ever.
That’s wonderful. I will play now. Thank you.