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Question Number 42154 by ajfour last updated on 19/Aug/18

Commented by ajfour last updated on 19/Aug/18

$$Ifr<\mathit{\rho }<Rwherermeans$$$$inradius,andRthecirumradius$$$$of△ABC,thenlocatecentreS$$$$ofgivencircleinsidetriangle$$$$(usinga,b,c,and\rho )$$$$formaximumoverlapareaof$$$$circleand△ABC.$$

Commented by MJS last updated on 19/Aug/18

$$\mathrm{it}\mathrm{seems}\mathrm{obvious}\mathrm{that}\mathrm{with}\rho ⩽R\mathrm{we}\mathrm{get}$$$$\rho =R,\mathrm{with}\rho <R\mathrm{we}\mathrm{get}\rho =R-\mathrm{\Delta }\mathrm{with}$$$$\mathrm{\Delta }\to 0...$$

Commented by ajfour last updated on 19/Aug/18

$$willSbethecircumcentreorsome$$$$otherpoint,israthermyquestion$$$$Sir?$$

Commented by MJS last updated on 19/Aug/18

$$\mathrm{because}\mathrm{the}\mathrm{greatest}\mathrm{possible}\mathrm{overlapping}$$$$\mathrm{area}\mathrm{is}\mathrm{the}\mathrm{triangle}\mathrm{itself},\mathrm{the}\mathrm{circle}\mathrm{is}\mathrm{the}$$$$\mathrm{circumcircle}\mathrm{so}\mathrm{the}\mathrm{center}\mathrm{is}\mathrm{the}\mathrm{circumcenter}$$$$$$$$\mathrm{remember}\mathrm{the}\mathrm{circumcircle}\mathrm{is}\mathrm{the}\mathrm{smallest}$$$$\mathrm{possible}\mathrm{circle}\mathrm{including}\mathrm{the}\mathrm{triangle}\iff$$$$\mathrm{the}\mathrm{smallest}\mathrm{circle}\mathrm{with}\mathrm{any}\mathrm{other}\mathrm{center},$$$$\mathrm{including}\mathrm{the}\mathrm{triangle}\mathrm{is}\mathrm{greater}\mathrm{than}\mathrm{the}$$$$\mathrm{circumcircle}$$

Commented by ajfour last updated on 19/Aug/18

$$thanksfortheexplanationSir.$$

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