Can-we-expand-the-Descartes-Theorem-to-more-than-4-circles-

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

$${Can}\:{we}\:{expand}\:{the}\:{Descartes}'\:{Theorem}\:{to}\:{more}\:{than}\:\mathrm{4}\:{circles}? \\ $$

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

such that two circles touch each other, their radii can be any values. such that three circles touch each other, their radii can be any values. such that four circles touch each other, their radii must satisfy a certain relation. this is the Descartes′ theorem. such that 5 (or more) circles touch each other, this is not possible!

$${such}\:{that}\:\boldsymbol{{two}}\:{circles}\:{touch}\:{each} \\ $$$${other},\:{their}\:{radii}\:{can}\:{be}\:{any}\:{values}. \\ $$$$ \\ $$$${such}\:{that}\:\boldsymbol{{three}}\:{circles}\:{touch}\:{each} \\ $$$${other},\:{their}\:{radii}\:{can}\:{be}\:{any}\:{values}. \\ $$$$ \\ $$$${such}\:{that}\:\boldsymbol{{four}}\:{circles}\:{touch}\:{each} \\ $$$${other},\:{their}\:{radii}\:{must}\:{satisfy}\:{a} \\ $$$${certain}\:{relation}.\:{this}\:{is}\:{the} \\ $$$${Descartes}'\:{theorem}. \\ $$$$ \\ $$$${such}\:{that}\:\mathrm{5}\:\left(\boldsymbol{{or}}\:\boldsymbol{{more}}\right)\:{circles}\:{touch} \\ $$$${each}\:{other},\:\boldsymbol{{this}}\:\boldsymbol{{is}}\:\boldsymbol{{not}}\:\boldsymbol{{possible}}! \\ $$

Commented by GDVilla last updated on 24/Dec/24

$$\mathrm{I}\:\mathrm{think}\:\mathrm{not} \\ $$$$ \\ $$

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

it′s just impossible that there exist 5 (or more) circles which touch each other!

$${it}'{s}\:{just}\:{impossible}\:{that}\:{there}\:{exist} \\ $$$$\mathrm{5}\:\left({or}\:{more}\right)\:{circles}\:{which}\:{touch}\:{each} \\ $$$${other}! \\ $$

Commented by TonyCWX08 last updated on 25/Dec/24

Commented by TonyCWX08 last updated on 25/Dec/24

$${What}\:{about}\:{this}? \\ $$

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

it doesn′t show that each circle touches every other circles. again, there are at most 4 circles which can touch each other.

$${it}\:{doesn}'{t}\:{show}\:{that}\:{each}\:{circle}\: \\ $$$${touches}\:{every}\:{other}\:{circles}.\:{again}, \\ $$$${there}\:{are}\:{at}\:{most}\:\mathrm{4}\:{circles}\:{which} \\ $$$${can}\:{touch}\:{each}\:{other}.\: \\ $$

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

$${these}\:{are}\:{examples}\:{that}\:{all}\:{circles} \\ $$$${touch}\:{each}\:{other}: \\ $$

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

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

$${this}\:{is}\:{not}\:{an}\:{example}\:{that}\:{all}\: \\ $$$${circles}\:{touch}\:{each}\:{other}: \\ $$

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

Commented by TonyCWX08 last updated on 25/Dec/24

$${Okay}\:{sir}. \\ $$

Commented by TonyCWX08 last updated on 25/Dec/24

$${Can}\:{you}\:{solve}\:{for}\:{the}\:{area}\:{of}\:{shaded}\:{region}? \\ $$

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

a series of circles inscribed between two circles like following example are a so−called Steiner chain.

$${a}\:{series}\:{of}\:{circles}\:{inscribed}\:{between} \\ $$$${two}\:{circles}\:{like}\:{following}\:{example} \\ $$$${are}\:{a}\:{so}−{called}\:\boldsymbol{{Steiner}}\:\boldsymbol{{chain}}. \\ $$

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