Question Number 70587 by behi83417@gmail.com last updated on 05/Oct/19
Commented by behi83417@gmail.com last updated on 05/Oct/19
$$\mathrm{ABCD},\mathrm{is}\:\mathrm{given}. \\ $$$$\mathrm{AB}=\boldsymbol{\mathrm{a}},\mathrm{BC}=\boldsymbol{\mathrm{b}},\mathrm{CD}=\boldsymbol{\mathrm{c}},\mathrm{DA}=\boldsymbol{\mathrm{d}},\mathrm{AC}=\boldsymbol{\mathrm{m}},\mathrm{BD}=\boldsymbol{\mathrm{n}}. \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}: \\ $$$$\:\:\:\:\:\left(\boldsymbol{\mathrm{mn}}\right)^{\mathrm{2}} =\left(\boldsymbol{\mathrm{ac}}\right)^{\mathrm{2}} +\left(\boldsymbol{\mathrm{bd}}\right)^{\mathrm{2}} −\mathrm{2}\boldsymbol{\mathrm{abcd}}.\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{C}}\right) \\ $$