Question Number 199774 by Mingma last updated on 09/Nov/23
Answered by ajfour last updated on 09/Nov/23
Commented by ajfour last updated on 09/Nov/23
$${P}\:\left({perimeter}\:{of}\:\bigtriangleup\:{ABC}\right) \\ $$$$\:\:=\:\mathrm{2}{R}+\mathrm{2}{R}\mathrm{cos}\:\theta \\ $$$${left}\:{part}\:{of}\:{it}\:{cut}\:{by}\:{MN} \\ $$$$\:=\:{BM}+{BN}\left(={AB}\right) \\ $$$$\:\:={R}\mathrm{cos}\:\theta+{R}\:\:=\frac{{P}}{\mathrm{2}} \\ $$$$\:{S}\left(\:{Area}\:\bigtriangleup{ABC}\right)=\:{R}^{\mathrm{2}} \mathrm{sin}\:\theta\mathrm{cos}\:\theta \\ $$$$\:\:{area}\:{cut}\:{by}\:{MN}\:=\:\frac{{BM}}{\mathrm{2}}×{MN} \\ $$$$\:\:\:\:\:\:\:\:=\:\frac{{R}\mathrm{cos}\:\theta}{\mathrm{2}}×{R}\mathrm{sin}\:\theta\:=\:\frac{{S}}{\mathrm{2}} \\ $$
Commented by Mingma last updated on 09/Nov/23
Very elegant solution!