The equations of the sides AB, BC and CA of a triangle ABC are 2x +...

We have,

Since, $G\left(2, a\right)$, so

$\dfrac{1 + \alpha + \beta + 3}{3} = 2$ and $\dfrac{- 2 - 2\alpha + \beta}{3} = a$

$⇒ \alpha + \beta = 2$ and $- 2\alpha + \beta = 3a + 2$

So, $\alpha = - a, \beta = 2 + a$

So, $B ≡ \left(- a, 2a\right), C ≡ \left(5 + a, 2 + a\right)$

Now, $B$ and $C$ lies on the line $x + py = 21a$, so

$- a + 2pa = 21a$

$⇒ p = 11$

Also,

$5 + a + 11\left(2 + a\right) = 21a$

$⇒ 27 = 9a ⇒ a = 3$

So,

$B ≡ \left(- 3, 6\right), C ≡ \left(8, 5\right)$

So,

$B{C}^{2} = 121 + 1 = 122$