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.


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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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