Soddy

From: philippe 92 (antispam_at_free.invalid)
Date: 10/21/04 

Date: Thu, 21 Oct 2004 11:51:00 +0200

Hello,

On the Mathworld page about isoperimetric point, I read :
( http://mathworld.wolfram.com/IsoperimetricPoint.html )

The isoperimetric point exists iff the largest angle of the
triangle satisfies :
max(A,B,C) < 2 Arcsin(4/5) = 106.26... deg
or equivalently a + b + c > 4R + r

When the isoperimetric point exists, it is the outer Soddy center.

Let's take an example
BC = a = 325 = 100 + 225
AC = b = 261 = 36 + 225
AB = c = 136 = 36 + 100

The three mutually tangent circles (A), (B), (C) have radii
rA = 36, rB = 100, rC = 225.

The radii of the Soddy circles are solutions of :
( http://mathworld.wolfram.com/SoddyCircles.html )
2(1/rA^2 + 1/rB^2 + 1/rC^2 + 1/x^2) = ((1/rA + 1/rB + 1/rC + 1/x)^2

x = 225/19 = 11.8421...
and 1/x = 0
the outer Soddy circle is a straight line !
hence there can't be an isoperimetric point (rejected to
infinity).

Nevertheless angle A is cos(A) = (b^2 + c^2 - a^2)/(2bc) gives
A = 105.53... deg < 106.26

????

Compute r from S = pr = sqrt(p.rA.rB.rC)
p = (a+b+c)/2 = 361
r = sqrt(rA.rB.rC/p) = 900/19 = 47.368421...
and S = 17100
R = abc/(4S) = 6409/38 = 168.65789...

a + b + c = 722
4R + r = 13718/19 = 722
That last iff condition seems OK but the max angle condition
seems to be false.

The angle iff condition would mean the wrong "theorem" :

Let 3 circles, mutually tangent externally and tangent to a
common line. The angle of centers is allways 106.26... deg.

Any opinion or point out my mistake ? 

-- 
philippe
(chephip at free dot fr)

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