Question Number 165722 by mr W last updated on 06/Feb/22
Commented by mr W last updated on 08/Feb/22
![Find the maximum area of a triangle whose vertices lie on three circles with radii a, b, c respectively which touch each other. [Q36270 reposted]](Q165723.png)
$${Find}\:{the}\:{maximum}\:{area}\:{of}\:{a}\:{triangle} \\ $$$${whose}\:{vertices}\:{lie}\:{on}\:{three}\:{circles} \\ $$$${with}\:{radii}\:{a},\:{b},\:{c}\:{respectively}\:{which} \\ $$$${touch}\:{each}\:{other}. \\ $$$$ \\ $$$$\left[\boldsymbol{{Q}}\mathrm{36270}\:\boldsymbol{{reposted}}\right] \\ $$
Answered by aleks041103 last updated on 07/Feb/22
$${for}\:{a}\:{fixed}\:{p}\:{and}\:{q}\:{the}\:{area}\:{is}\:{max} \\ $$$${when}\:{the}\:{height}\:{is}\:{max}.\:{this}\:{is}\:{achieved} \\ $$$${when}\:{RC}\bot{PQ}. \\ $$$${Analogously}\:{for}\:{the}\:{other}\:{sides} \\ $$$$\:{and}\:{therefore} \\ $$$${PA}\bot{RQ} \\ $$$${QB}\bot{PR} \\ $$$${RC}\bot{PQ} \\ $$
Commented by mr W last updated on 08/Feb/22
$${yes}\:{sir}!\:{just}\:{as}\:{i}\:{also}\:{said}\:{in}\:{Q}\mathrm{36270}. \\ $$$${but}\:{we}\:{still}\:{haven}'{t}\:{got}\:{a}\:{satisfactory} \\ $$$${solution}\:{with}\:{concrete}\:{result}.\:{please} \\ $$$${also}\:{give}\:{a}\:{try}\:{sir}! \\ $$