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Question Number 46453 by MrW3 last updated on 26/Oct/18

Commented by MrW3 last updated on 27/Oct/18

$$Threecircleswithradiia,bandc$$$$toucheachother.Theircircumscribed$$$$circlehastheradiusRandtheir$$$$inscribedcirclehastheradiusr.$$$$Provethat$$$$R=\frac{1}{2\sqrt{\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}}-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}}$$$$r=\frac{1}{2\sqrt{\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$

Commented by behi83417@gmail.com last updated on 26/Oct/18

$$vetybeautifulquestion!$$

Commented by MJS last updated on 26/Oct/18

$$\mathrm{this}\mathrm{is}\mathrm{related}\mathrm{to}\mathrm{a}\mathrm{former}\mathrm{question},{\mathrm{couldn}}^{\prime }\mathrm{t}$$$$\mathrm{find}\mathrm{it}\mathrm{right}\mathrm{now}.$$$$\mathrm{the}\mathrm{inscribed}\mathrm{circle}\mathrm{always}\mathrm{exists}\mathrm{but}\mathrm{the}$$$$\mathrm{circumcircle}\mathrm{sometimes}\mathrm{degenerates}\mathrm{to}\mathrm{a}$$$$\mathrm{common}\mathrm{tangent}\mathrm{or}\mathrm{even}\mathrm{a}\mathrm{circle}\mathrm{touching}$$$$\mathrm{the}3\mathrm{given}\mathrm{from}\mathrm{outside}(\mathrm{imagine}2\mathrm{big}\mathrm{circles}$$$$\mathrm{and}\mathrm{the}{3}^{\mathrm{rd}}\mathrm{one}\mathrm{small}\mathrm{enough}\mathrm{to}\mathrm{fit}\mathrm{in}\mathrm{the}$$$$‘‘{\mathrm{gap}}^{″}\mathrm{between}\mathrm{them})$$

Commented by MJS last updated on 26/Oct/18

$$\mathrm{I}\mathrm{once}\mathrm{downloaded}\mathrm{my}\mathrm{solution}\mathrm{as}\mathrm{an}\mathrm{image},$$$$\mathrm{here}\mathrm{it}\mathrm{is}(\mathrm{sorry}{\mathrm{can}}^{\prime }\mathrm{t}\mathrm{change}\mathrm{it}\mathrm{so}\mathrm{it}{\mathrm{doesn}}^{\prime }\mathrm{t}$$$$\mathrm{fit}\mathrm{exactly}\mathrm{here}\mathrm{but}\mathrm{it}\mathrm{might}\mathrm{help})$$

Commented by MJS last updated on 26/Oct/18

Commented by MrW3 last updated on 26/Oct/18

$$Thisquestionisindeedderivedfrom$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$ThegivenformulasforRandrare$$$$simple,butnoteasytoprove.Canwe$$$$findaneasywayfortheproof?$$

Answered by ajfour last updated on 27/Oct/18

$$\mathrm{\Sigma }\sqrt{R[R-(a+b)]ab}=\sqrt{(a+b+c)abc}$$$$cantsolveforR!$$

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