If the coordinates of the vertices of an equilateral triangle with sides of length 'a' are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ then \[{\begin{vmatrix}x_1 & y_1 &1\\x_2 & y_2 &1\\x_3 & y_3 & 1\end{vmatrix}}^2=\frac{3a^4}{4}.\]

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