Can we expand the Descartes′ Theorem to more than 4 circles?


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Question Number 214936 by TonyCWX08 last updated on 24/Dec/24

$C a n w e e x p a n d t h e {D e s c a r t e s}^{'} T h e o r e m t o m o r e t h a n 4 c i r c l e s ?$
Commented by mr W last updated on 25/Dec/24

$s u c h t h a t t w o c i r c l e s t o u c h e a c h$ $o t h e r, t h e i r r a d i i c a n b e a n y v a l u e s .$ $s u c h t h a t t h r e e c i r c l e s t o u c h e a c h$ $o t h e r, t h e i r r a d i i c a n b e a n y v a l u e s .$ $s u c h t h a t f o u r c i r c l e s t o u c h e a c h$ $o t h e r, t h e i r r a d i i m u s t s a t i s f y a$ $c e r t a i n r e l a t i o n . t h i s i s t h e$ ${D e s c a r t e s}^{'} t h e o r e m .$ $s u c h t h a t 5 (o r m o r e) c i r c l e s t o u c h$ $e a c h o t h e r, t h i s i s n o t p o s s i b l e!$
Commented by GDVilla last updated on 24/Dec/24

$I think not$
Commented by mr W last updated on 25/Dec/24

${i t}^{'} s j u s t i m p o s s i b l e t h a t t h e r e e x i s t$ $5 (o r m o r e) c i r c l e s w h i c h t o u c h e a c h$ $o t h e r!$
Commented by TonyCWX08 last updated on 25/Dec/24

Commented by TonyCWX08 last updated on 25/Dec/24

$W h a t a b o u t t h i s ?$
Commented by mr W last updated on 25/Dec/24

$i t {d o e s n}^{'} t s h o w t h a t e a c h c i r c l e$ $t o u c h e s e v e r y o t h e r c i r c l e s . a g a i n,$ $t h e r e a r e a t m o s t 4 c i r c l e s w h i c h$ $c a n t o u c h e a c h o t h e r .$
Commented by mr W last updated on 25/Dec/24

$t h e s e a r e e x a m p l e s t h a t a l l c i r c l e s$ $t o u c h e a c h o t h e r :$
Commented by mr W last updated on 25/Dec/24

Commented by mr W last updated on 25/Dec/24

$t h i s i s n o t a n e x a m p l e t h a t a l l$ $c i r c l e s t o u c h e a c h o t h e r :$
Commented by mr W last updated on 25/Dec/24

Commented by TonyCWX08 last updated on 25/Dec/24

$O k a y s i r .$
Commented by TonyCWX08 last updated on 25/Dec/24

$C a n y o u s o l v e f o r t h e a r e a o f s h a d e d r e g i o n ?$
Commented by mr W last updated on 25/Dec/24

$a s e r i e s o f c i r c l e s i n s c r i b e d b e t w e e n$ $t w o c i r c l e s l i k e f o l l o w i n g e x a m p l e$ $a r e a s o - c a l l e d S t e i n e r c h a i n .$
Commented by mr W last updated on 25/Dec/24

Commented by TonyCWX08 last updated on 25/Dec/24

$O k a y .$ $T h a n k s .$ $I^{'} l l d e a l i t m y s e l f .$
Commented by mr W last updated on 25/Dec/24

$t e l l m e y o u r r e s u l t w h e n {y o u}^{'} v e g o t$ $i t .$
Commented by mr W last updated on 31/Dec/24

$s e e Q 215195$
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