Question Number 42154 by ajfour last updated on 19/Aug/18
Commented by ajfour last updated on 19/Aug/18
$${If}\:\:{r}\:<\:\boldsymbol{\rho}\:<{R}\:\:{where}\:{r}\:{means} \\ $$$${inradius},\:{and}\:{R}\:{the}\:{cirumradius} \\ $$$${of}\:\bigtriangleup{ABC},\:{then}\:{locate}\:{centre}\:{S} \\ $$$${of}\:{given}\:{circle}\:{inside}\:{triangle} \\ $$$$\left({using}\:{a},{b},{c},\:{and}\:\rho\right) \\ $$$${for}\:{maximum}\:{overlap}\:{area}\:{of} \\ $$$${circle}\:{and}\:\bigtriangleup{ABC}. \\ $$
Commented by MJS last updated on 19/Aug/18
$$\mathrm{it}\:\mathrm{seems}\:\mathrm{obvious}\:\mathrm{that}\:\mathrm{with}\:\rho\leqslant{R}\:\mathrm{we}\:\mathrm{get} \\ $$$$\rho={R},\:\mathrm{with}\:\rho<{R}\:\mathrm{we}\:\mathrm{get}\:\rho={R}−\Delta\:\mathrm{with} \\ $$$$\Delta\rightarrow\mathrm{0}... \\ $$
Commented by ajfour last updated on 19/Aug/18
$${will}\:{S}\:{be}\:{the}\:{circumcentre}\:{or}\:{some} \\ $$$${other}\:{point},\:{is}\:{rather}\:{my}\:{question} \\ $$$${Sir}\:? \\ $$
Commented by MJS last updated on 19/Aug/18
$$\mathrm{because}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{possible}\:\mathrm{overlapping} \\ $$$$\mathrm{area}\:\mathrm{is}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{itself},\:\mathrm{the}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{circumcircle}\:\mathrm{so}\:\mathrm{the}\:\mathrm{center}\:\mathrm{is}\:\mathrm{the}\:\mathrm{circumcenter} \\ $$$$ \\ $$$$\mathrm{remember}\:\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{is}\:\mathrm{the}\:\mathrm{smallest} \\ $$$$\mathrm{possible}\:\mathrm{circle}\:\mathrm{including}\:\mathrm{the}\:\mathrm{triangle}\:\Leftrightarrow \\ $$$$\mathrm{the}\:\mathrm{smallest}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{any}\:\mathrm{other}\:\mathrm{center}, \\ $$$$\mathrm{including}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{is}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{the} \\ $$$$\mathrm{circumcircle} \\ $$
Commented by ajfour last updated on 19/Aug/18
$${thanks}\:{for}\:{the}\:{explanation}\:{Sir}. \\ $$