a-b-c-a-b-1-2-a-b-c-b-c-1-2-a-b-c-a-c-1-a-b-c-

Question Number 10067 by konen last updated on 22/Jan/17

\frac{a \times b \times c}{a + b} = - \frac{1}{2}

\frac{a \times b \times c}{b + c} = \frac{1}{2}

\frac{a \times b \times c}{a + c} = 1

\Rightarrow a \times b \times c = ?

Answered by mrW1 last updated on 22/Jan/17

l e t u = a \times b \times c

\Rightarrow a + b = - 2 u \dots (i)

\Rightarrow b + c = 2 u \dots (i i)

\Rightarrow a + c = u \dots (i i i)

(i i) - (i i i) :

b - a = u \dots (i v)

(i) + (i v) :

2 b = - u

b = - \frac{u}{2}

c = 2 u - b = 2 u + \frac{u}{2} = \frac{5 u}{2}

a = - 2 u - b = - 2 u + \frac{u}{2} = - \frac{3 u}{2}

u = a \times b \times c = (\frac{- 3 u}{2}) \times (\frac{- u}{2}) \times (\frac{5 u}{2}) = \frac{15}{8} u^{3}

(\frac{15}{8} u^{2} - 1) u = 0

u \neq 0

\Rightarrow \frac{15}{8} u^{2} - 1 = 0

u^{2} = \frac{8}{15}

\Rightarrow a \times b \times c = u = \pm \sqrt{\frac{8}{15}} = \pm 2 \sqrt{\frac{2}{15}}

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