# Express in terms in terms of $a$

`I have a terms written below

$((x_1-x_2)(y_2-y_3)-(x_2-x_3)(y_1-y_2))^2$

Also i am given vertices of equilateral triangle with side $a$ and vertices as $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. I am to write above term in terms of $a$.I tried using distance formula for equilateral triangle and then to eliminate $x_i$ in original term , but it didnot work out.

Any suggestions

Thanks

• Your question is not enough clear. – Kanwaljit Singh Feb 5 '17 at 12:25
• It is not a question nor an equation. – mathreadler Feb 5 '17 at 12:26

Area of triangle with vertices $(x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})$ is given by
$\triangle=\dfrac{1}{2}\begin{vmatrix}x_{1}&y_{1}&1\\x_{2}&y_{2}&1\\x_{3}&y_{3}&1\end{vmatrix}$
$=\dfrac{1}{2}\begin{vmatrix}x_{1}-x_{2}&y_{1}-y_{2}&0\\x_{2}-x_{3}&y_{2}-y_{3}&0\\x_{3}&y_{3}&1\end{vmatrix}=\dfrac{1}{2}\left[(x_{1}-x_{2})(y_{2}-y_{3})-(x_{2}-x_{3})(y_{1}-y_{2})\right]$
Thus,$\left((x_{1}-x_{2})(y_{2}-y_{3})-(x_{2}-x_{3})(y_{1}-y_{2})\right)^2=4\times$ $(\triangle)^2=4\times\left(\dfrac{\sqrt{3}}{4}a^2\right)^2=\dfrac{3}{4}a^4$