Question Number 84335 by ajfour last updated on 11/Mar/20
Commented by ajfour last updated on 11/Mar/20
$${min}\left({p}+{q}+{r}\right)={f}\left({a},{b},{c}\right)\:.\:{Find}\:{f}. \\ $$
Commented by mr W last updated on 12/Mar/20
$$\Delta={area}\:{of}\:{ABC}=\frac{\sqrt{\left({a}+{b}+{c}\right)\left(−{a}+{b}+{c}\right)\left({a}−{b}+{c}\right)\left({a}+{b}−{c}\right)}}{\mathrm{4}} \\ $$$${min}\left({p}+{q}+{r}\right)=\mathrm{3}{R}=\frac{\mathrm{3}{abc}}{\mathrm{4}\Delta} \\ $$$$=\frac{\mathrm{3}{abc}}{\:\sqrt{\left({a}+{b}+{c}\right)\left(−{a}+{b}+{c}\right)\left({a}−{b}+{c}\right)\left({a}+{b}−{c}\right)}} \\ $$
Commented by ajfour last updated on 12/Mar/20
$${does}\:{that}\:{mean},\: \\ $$$${p}+{q}+{r}\:{is}\:{minimum}\:{when} \\ $$$${p}={q}={r}={R}\:,\:{Sir}\:? \\ $$
Commented by mr W last updated on 12/Mar/20
$${yes}.\:{due}\:{to}\:{symmetry}. \\ $$