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Question-42154




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

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