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Question-104411




Question Number 104411 by ajfour last updated on 21/Jul/20
Commented by ajfour last updated on 21/Jul/20
Relate a,b,c (when they r real).
$${Relate}\:\boldsymbol{{a}},\boldsymbol{{b}},\boldsymbol{{c}}\:\left({when}\:{they}\:{r}\:{real}\right). \\ $$
Commented by mr W last updated on 21/Jul/20
a+b=((b+c)/2)  c+a=(((√3)(b+c))/2)  with α=(a/c),β=(b/c)  ⇒2α+β=1  ⇒−2α+(√3)β=2−(√3)  ⇒α=2−(√3)≈0.268  ⇒β=2(√3)−3≈0.464
$${a}+{b}=\frac{{b}+{c}}{\mathrm{2}} \\ $$$${c}+{a}=\frac{\sqrt{\mathrm{3}}\left({b}+{c}\right)}{\mathrm{2}} \\ $$$${with}\:\alpha=\frac{{a}}{{c}},\beta=\frac{{b}}{{c}} \\ $$$$\Rightarrow\mathrm{2}\alpha+\beta=\mathrm{1} \\ $$$$\Rightarrow−\mathrm{2}\alpha+\sqrt{\mathrm{3}}\beta=\mathrm{2}−\sqrt{\mathrm{3}} \\ $$$$\Rightarrow\alpha=\mathrm{2}−\sqrt{\mathrm{3}}\approx\mathrm{0}.\mathrm{268} \\ $$$$\Rightarrow\beta=\mathrm{2}\sqrt{\mathrm{3}}−\mathrm{3}\approx\mathrm{0}.\mathrm{464} \\ $$
Commented by ajfour last updated on 21/Jul/20
It isn′t like that Sir...  please view Q.104445
$${It}\:{isn}'{t}\:{like}\:{that}\:{Sir}… \\ $$$${please}\:{view}\:{Q}.\mathrm{104445} \\ $$
Commented by mr W last updated on 21/Jul/20
you are right sir! what i showed is  only a special case.  generally we have:  (a+c)^2 =(a+b)^2 +(b+c)^2 −(a+b)(b+c)  ⇒b^2 +(a+c)b−3ac=0  or  ⇒c=(((a+b)b)/(3a−b))
$${you}\:{are}\:{right}\:{sir}!\:{what}\:{i}\:{showed}\:{is} \\ $$$${only}\:{a}\:{special}\:{case}. \\ $$$${generally}\:{we}\:{have}: \\ $$$$\left({a}+{c}\right)^{\mathrm{2}} =\left({a}+{b}\right)^{\mathrm{2}} +\left({b}+{c}\right)^{\mathrm{2}} −\left({a}+{b}\right)\left({b}+{c}\right) \\ $$$$\Rightarrow{b}^{\mathrm{2}} +\left({a}+{c}\right){b}−\mathrm{3}{ac}=\mathrm{0} \\ $$$${or} \\ $$$$\Rightarrow{c}=\frac{\left({a}+{b}\right){b}}{\mathrm{3}{a}−{b}} \\ $$
Commented by ajfour last updated on 21/Jul/20
Excellent and very pragmatic notice  Sir, i follow it well, thanks a lot.
$${Excellent}\:{and}\:{very}\:{pragmatic}\:{notice} \\ $$$${Sir},\:{i}\:{follow}\:{it}\:{well},\:{thanks}\:{a}\:{lot}. \\ $$

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