If-you-know-b-2-c-2-a-2-2bc-2-c-2-a-2-b-2-2ca-2-a-2-b-2-c-2-2ab-2-3-then-what-s-the-value-of-b-2-c-2-a-2-2bc-c-2-a-2-b-2-2ac-a-2-b-2-c-2-2

Question Number 84871 by tw000001 last updated on 17/Mar/20

If you know

{(\frac{b^{2} + c^{2} - a^{2}}{2 b c})}^{2} + {(\frac{c^{2} + a^{2} - b^{2}}{2 c a})}^{2} + {(\frac{a^{2} + b^{2} - c^{2}}{2 a b})}^{2} = 3,

then {what}^{'} s the value of

\frac{b^{2} + c^{2} - a^{2}}{2 b c} + \frac{c^{2} + a^{2} - b^{2}}{2 a c} + \frac{a^{2} + b^{2} - c^{2}}{2 a b} ?

Commented by ajfour last updated on 17/Mar/20

1 ?

Answered by MJS last updated on 17/Mar/20

{(\frac{b^{2} + c^{2} - a^{2}}{2 b c})}^{2} + {(\frac{c^{2} + a^{2} - b^{2}}{2 c a})}^{2} + {(\frac{a^{2} + b^{2} - c^{2}}{2 a b})}^{2} = 3

\Leftrightarrow

(a^{2} + b^{2} + c^{2}) (a + b + c) (- a + b + c) (a - b + c) (a + b - c) = 0

\Leftrightarrow

c = - a - b \lor c = - a + b \lor c = a - b \lor c = a + b

\Rightarrow

\frac{b^{2} + c^{2} - a^{2}}{2 b c} + \frac{c^{2} + a^{2} - b^{2}}{2 c a} + \frac{a^{2} + b^{2} - c^{2}}{2 a b} = x

x = - 3 \lor x = 1

the red c gives - 3, the others give 1

Commented by MJS last updated on 17/Mar/20

{you}^{'} re welcome

Commented by tw000001 last updated on 17/Mar/20

Thank you for your solution .