If `A(x_1, y_1),B(x_2, y_2)a n dC(x_3,y_3)` are vertices of an equilateral triangle whose each side is equal to `a` , then prove that `|x_1y_1 2x_2y_2 2x_3y_3 2|^2=3a^4`

Explanation

We have given : `A(x_1, y_1),B(x_2, y_2)a n dC(x_3,y_3)`
are vertices of an equilateral triangle whose each side is equal to `a`

Then we have to prove that
`|(x_1,y_1, 2),(x_2,y_2, 2),(x_3,y_3 ,2)|^2=3a^4`

Now as we know that the area of tiangle with given vertices can given by

`=1/2[(x_1,y_1, 2),(x_2,y_2, 2),(x_3,y_3 ,2)]`.....(i)

And area of equilateral triangle with side `a` is `(sqrt3)/(4)a^2......(ii)`

Then from (i) and (ii) we get

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