Question Number 14128 by chux last updated on 28/May/17
$$\mathrm{If}\:\:\frac{\mathrm{P}{x}}{\left({b}−{c}\right)}\:=\:\frac{\mathrm{Q}{y}}{\left({c}−{a}\right)}\:=\:\frac{\mathrm{R}{z}}{\left({a}−{b}\right)}\:,\:\mathrm{then}\:\mathrm{find} \\ $$$$\mathrm{P}{ax}\:+\:\mathrm{Q}{by}\:+\:\mathrm{R}{cz}. \\ $$
Answered by ajfour last updated on 28/May/17
$${let}\:\:\frac{{Px}}{{b}−{c}}=\frac{{Qy}}{{c}−{a}}=\frac{{Rz}}{{a}−{b}}={m}\:\:\left({say}\right) \\ $$$${to}\:{find}:\:\:\: \\ $$$${Pax}+{Qby}+{Rcz}= \\ $$$$\:\:{ma}\left({b}−{c}\right)+{mb}\left({c}−{a}\right)+{mc}\left({a}−{b}\right) \\ $$$$\:\:\:\:\:={m}\left({ab}−{ac}+{bc}−{ab}+{ac}−{bc}\right) \\ $$$$\:\:\:\:\:=\mathrm{0}\:. \\ $$
Commented by Joseph Paul Byaraujo. last updated on 29/May/17
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