Question Number 56090 by Tawa1 last updated on 10/Mar/19
Commented by math1967 last updated on 10/Mar/19
$${pls}\:{check}\:{I}\:{think}\:{it}\:{should}\:{be} \\ $$$$\begin{vmatrix}{{bc}}&{{ca}}&{{ab}}\\{{a}}&{{b}}&{{c}}\\{{a}^{\mathrm{2}} }&{{b}^{\mathrm{2}} }&{{c}^{\mathrm{2}} }\end{vmatrix} \\ $$
Commented by Tawa1 last updated on 10/Mar/19
$$\mathrm{Please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{do}\:\mathrm{it}\:\mathrm{like}\:\mathrm{this}\: \\ $$
Commented by Tawa1 last updated on 10/Mar/19
$$\mathrm{But}\:\mathrm{that}\:\mathrm{is}\:\mathrm{the}\:\mathrm{question}. \\ $$
Commented by math1967 last updated on 10/Mar/19
$$\begin{vmatrix}{{bc}−{ca}}&{{ca}−{ab}}&{{ab}}\\{{a}−{b}}&{{b}−{c}}&{{c}}\\{{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }&{{b}^{\mathrm{2}} −{c}^{\mathrm{2}\:} }&{{c}^{\mathrm{2}\:} \:\:}\end{vmatrix}{c}_{\mathrm{1}\:\:} −{c}_{\mathrm{2}} ,{c}_{\mathrm{2}\:} −{c}_{\mathrm{3}\:} \\ $$$$\left({a}−{b}\right)\left({b}−{c}\right)\begin{vmatrix}{−{c}}&{−{a}}&{{ab}}\\{\mathrm{1}}&{\mathrm{1}}&{{c}}\\{{a}+{b}}&{{b}+{c}}&{{c}^{\mathrm{2}} }\end{vmatrix} \\ $$$$=\left({a}−{b}\right)\left({b}−{c}\right)\begin{vmatrix}{−{c}}&{{c}−{a}}&{{ab}}\\{\mathrm{1}}&{\mathrm{0}}&{{c}}\\{{a}+{b}\:}&{{c}−{a}}&{{c}^{\mathrm{2}} }\end{vmatrix}{c}_{\mathrm{2}} −{c}_{\mathrm{1}} \\ $$$$=\left({a}−{b}\right)\left({b}−{c}\right)\left({c}−{a}\right)\begin{vmatrix}{−{c}}&{\mathrm{1}}&{{ab}}\\{\mathrm{1}}&{\mathrm{0}}&{{c}}\\{{a}+{b}}&{\mathrm{1}}&{{c}^{\mathrm{2}} }\end{vmatrix} \\ $$$$=\left({a}−{b}\right)\left({b}−{c}\right)\left({c}−{a}\right)\left\{{c}^{\mathrm{2}} −{c}^{\mathrm{2}} +{ca}+{bc}+{ab}\right\} \\ $$$$=\left({b}−{a}\right)\left({c}−{b}\right)\left({c}−{a}\right)\left({ab}+{bc}+{ca}\right) \\ $$
Commented by Tawa1 last updated on 11/Mar/19
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$