Re: Soddy
From: Ignacio Larrosa Cañestro (ilarrosaQUITARMAYUSCULAS_at_mundo-r.com)
Date: 10/24/04
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Date: Mon, 25 Oct 2004 00:34:42 +0200
En el mensaje:41778684.2080508@free.invalid,
philippe 92 <antispam@free.invalid> escribió:
> 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 ?
I think that the limit angle 2*Arcsin(4/5) stand only for isisceles
triangles. I said it to Eric a lot of time ago, related with the question of
if the equal detour point is unique or not, but without effect until now. I
send that message on 11/11/2001:
*******************************************
In the article "Equal Detour Point " it says:
"If ABC has no Angle > 2*ArcSin[4/5], then the point given by the above
Trilinear Coordinates is the unique equal detour point. Otherwise, the
Isoperimetric Point is also equal detour. "
and in the article "Isoperimetric Point":
"The isoperimetric point exists Iff the largest Angle of the triangle
satisfies max(ABC) < 2*ArcSin[4/5]"
The critical angle in both cases is that iff the triangle is isosceles.
In generall, the critical angle is
T=arccos( - (2v^3 + 3v^2 + 2v)/(2v^4 + 6v^3 + 9v^2 + 6v + 2)
or
T= pi/2 + arctg(v(2v^2 + 3v + 2)/(2(v + 1)^2(v^2 + v + 1)))
where v=sqrt(z/y), being z and y the distances from B and C to the
intouch point of side a.
For that critical angle, the outer Soddy circle is a straight line.
T go to pi/2 when v ---> 0 or infinity, and 2*ArcSin[4/5] is its
maximun, that occurs when v=1, i.e. y=z, b=c and the triangle is
isosceles.
If I'm not mistaken, of course ...
*************************************************
-- Saludos, Ignacio Larrosa Cañestro A Coruña (España) ilarrosaQUITARMAYUSCULAS@mundo-r.com
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