Four-spheres-with-radii-a-b-c-and-d-touch-each-other-Find-the-radii-of-their-circumscribed-sphere-R-and-their-inscribed-sphere-r-in-terms-of-a-b-c-and-d-

Question Number 46481 by MrW3 last updated on 27/Oct/18

F o u r s p h e r e s w i t h r a d i i a, b, c a n d d

t o u c h e a c h o t h e r . F i n d t h e r a d i i o f

t h e i r c i r c u m s c r i b e d s p h e r e (R) a n d

t h e i r i n s c r i b e d s p h e r e (r) i n t e r m s o f

a, b, c a n d d .

Commented by MJS last updated on 27/Oct/18

plenty of cases . 3 spheres similar to 3 circles :

3^{rd} one could be very small compared to the

2 others, then the 4^{th} is still possible but no

circumscribed sphere could exist, even not

any sphere or plain touching the 4 spheres

from the outside .

inscribed sphere {doesn}^{'} t exist in some cases

too . imagine three spheres : their centers

are in a plain; the 4^{th} one could exactly fit

into the gap between them .

Commented by MrW3 last updated on 27/Oct/18

t h a n k y o u s i r!

g e n e r a l l y y o u a r e r i g h t .

b u t o n e c a n t r e a t a t f i r s t t h e c a s e t h a t s u c h

a s p h e r e e x i s t s . f r o m t h e r e s u l t o n e c a n

t h e n s e e w h a t i s t h e c o n d i t i o n t h a t s u c h

a s p h e r e e x i s t s . {l e t}^{'} s c o m e b a c k t o t h e

q u e s t i o n w i t h t h r e e c i r c l e s . f r o m t h e

r e s u l t s

R = \frac{1}{2 \sqrt{\frac{1}{a b} + \frac{1}{b c} + \frac{1}{c a}} - (\frac{1}{a} + \frac{1}{b} + \frac{1}{c})}

r = \frac{1}{2 \sqrt{\frac{1}{a b} + \frac{1}{b c} + \frac{1}{c a}} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}}

w e c a n s e e t h a t t h e i n s c r i b e d c i r c l e

a l w a y s e x i t s s i n c e w e g e t a v a l u e f o r

r w i t h a n y v a l u e s o f a, b a n d c . b u t

t h e c i r c u m c i r c l e {d o e s n}^{'} t a l w a y s e x i s t .

i f 2 \sqrt{\frac{1}{a b} + \frac{1}{b c} + \frac{1}{c a}} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} t h e r e i s

n o c i r c u m c i r c l e a t a l l, s i n c e R = \infty . a n d

i f 2 \sqrt{\frac{1}{a b} + \frac{1}{b c} + \frac{1}{c a}} < \frac{1}{a} + \frac{1}{b} + \frac{1}{c} t h e r e i s

a “ {c i r c u m c i r c l e}^{″}, b u t i t t a n g e n t s t h e

t h r e e c i r c l e s f r o m o u t s i d e, s i n c e i n t h i s

c a s e t h e v a l u e o f R i s n e g a t i v e .

s o m y o p i n i o n i s t h a t w e s h o u l d s t a r t

w i t h t h e g e n e r a l c a s e a n d t h e n s e e

w h i c h e x c e p t i o n s t h e r e a r e .

Commented by MJS last updated on 27/Oct/18

{you}^{'} re right

we can start with a ⩾ b ⩾ c ⩾ d

put C_{a} = (\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}) C_{b} = (\begin{matrix} 0 \\ y_{b} \\ 0 \end{matrix}) C_{c} = (\begin{matrix} x_{c} \\ y_{c} \\ 0 \end{matrix})

(I^{'} m a coordinate - method - lover)

the results of the 3 - circle - problem fit here

C_{d} = (\begin{matrix} x_{d} \\ y_{d} \\ z_{d} \end{matrix})

and we must find out the borders for c and d

depending on a and b so that a circumsphere

exists .

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