Question Number 191498 by MATHEMATICSAM last updated on 24/Apr/23
$$\mathrm{If}\:\mathrm{A}\left(\frac{\mathrm{2}{c}}{{a}}\:,\:\frac{{c}}{{b}}\right),\:\mathrm{B}\left(\frac{{c}}{{a}}\:,\:\mathrm{0}\right)\:\mathrm{and}\:\mathrm{C}\left(\frac{\mathrm{1}\:+\:{c}}{{a}}\:,\:\frac{\mathrm{1}}{{b}}\right) \\ $$$$\mathrm{are}\:\mathrm{three}\:\mathrm{points},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that},\: \\ $$$$\mathrm{i}.\:\:\frac{\left(\mathrm{AB}\right)^{\mathrm{2}} \:+\:\left(\mathrm{BC}\right)^{\mathrm{2}} }{\left(\mathrm{CA}\right)^{\mathrm{2}} }\:=\:\frac{{c}^{\mathrm{2}} \:+\:\mathrm{1}}{\left({c}\:−\:\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\mathrm{ii}.\:\left(\mathrm{AB}\right)^{\mathrm{2}} \:+\:\left(\mathrm{BC}\right)^{\mathrm{2}} \:−\:\left(\mathrm{AC}\right)^{\mathrm{2}} \:=\:\frac{\mathrm{2}{c}\left({a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \right)}{{a}^{\mathrm{2}} {b}^{\mathrm{2}} } \\ $$