If `A(x_1, y_1),B(x_2, y_2) ` and `C(x_3,y_3)` are vertices of an equilateral triangle whose each side is equal to `a` , then prove that `|[x_1,y_1, 2],[x_2,y_2, 2],[x_3,y_3, 2]|^2=3a^4`

Explanation

Area of an equilateral triangle=`(sqrt3 a^2)/4 `

area= `1/2|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|`

`sqrt3 a^2/4 = 1/2 |(x_1, y_1,1),(x_2,y_2,1),(x_3,y_3,1)|`

`sqrt3 a^2 = 2|(x_1 , y_1, 1),(x_2, y_2, 1),(x_3, y_3,1)|`

`sqrt3 a^2 = |(x_1, y_1, 2),(x_2,y_2,2),(x_3,y_3,2)|`

`3a^4 = |(x_1, y_1, 2),(x_2,y_2, 2),(x_3,y_3,2)|^2`

hence proved