given spheres of radius 2,2,3,3 tangent to all other and a smaller sphere tangent to all 4 find its radius
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Last edited by MrPolite; Jan 27th, 2003 at 05:03 PM.
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Originally posted by myself One that has the same derivative at the point of tangency. Or, put it another way, both spheres share the same tangent line.
My fault, should have been "same tangent plane", not "same tangent line"... and, of course, the derivative should be the partial derivatives.
a,b,c,d are spheres with radii 3,3,2 and 2 respectively
first figure:
The distance between two spheres is the sum of their radii
the triangles c-d-e, b-a-e are isosceles with legs 2+e resp 3+e
second figure:
the line from c+(d-c)/2 to a+(b-a)/2 goes trough e, and is perpendicular with a-b and b-c
their planes are perpendicular (due to symetry, can't bother explain this right now)
third figure:
thus a-d can be put in a box with sides 2,3 and z.
x^2+y^2+z^2=spacediagonale
z=sqrt(25-9-4)=sqrt(12) = sqrt((2+e)^2-2^2)+sqrt((3+e)^2-3^2)
12=e(e+4)+e(e+6)+2sqrt(e(e+4)e(e+6))
e=6/11
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