Question Number 94144 by Tony Lin last updated on 17/May/20
$${There}\:{is}\:{a}\:{moving}\:{point}\:{P}\:{in}\:{a}\:{triangle} \\ $$$$\:{ABC}\:{of}\:{which}\:{sides}\:{are}\:{a},{b},{c}\:{and}\:{a}>{b}>{c} \\ $$$${find}\:{the}\:{minimum}\:{and}\:{maximum} \\ $$$${of}\:{AP}+{BP}+{CP} \\ $$
Commented by Tony Lin last updated on 17/May/20
$${thanks}\:{sir}\:{I}\:{almost}\:{forgot}\:{R}=\frac{{abc}}{\mathrm{4}\Delta} \\ $$
Commented by mr W last updated on 17/May/20
$${minimum}=\mathrm{3}{R}=\frac{\mathrm{3}{abc}}{\:\sqrt{\left({a}+{b}+{c}\right)\left(−{a}+{b}+{c}\right)\left({a}−{b}+{c}\right)\left({a}+{b}−{c}\right)}} \\ $$$${when}\:{P}\:{is}\:{the}\:{circumcenter}. \\ $$$$ \\ $$$${maximum}={a}+{b} \\ $$$${when}\:{P}\:{is}\:{the}\:{vertex}\:{C}. \\ $$