# How to Prove this? If a + b + c = 0, Then Show that, a^2 - bc = b^2 - ca = c^2 - ab.

Apr 3, 2018

See the proof below

#### Explanation:

If $a + b + c = 0$

Then,

$a = - b - c = - \left(b + c\right)$

Therefore,

${a}^{2} - b c = {\left(- \left(b + c\right)\right)}^{2} - b c = {\left(b + c\right)}^{2} - b c$

$= {b}^{2} + {c}^{2} + 2 b c - b c$

$= {b}^{2} + {c}^{2} + b c$

${b}^{2} - c a = {b}^{2} - c \left(- \left(b + c\right)\right) = {b}^{2} + {c}^{2} + b c$

${c}^{2} - a b = {c}^{2} - \left(- \left(b + c\right) b\right) = {c}^{2} + {b}^{2} + b c$

So,

${a}^{2} - b c = {b}^{2} - c a = {c}^{2} - a b$

$Q E D$