Equilateral Triangle and Algebra

It is a simple exercise in trigonometry to show that $x_1 x_2+x_1 x_3+x_2 x_3=3\bar{x}^2-\frac{a^2}{4}$ for the vertices of any equilateral triangle, where $(\bar{x},\bar{y})$ is the centroid and $a$ is the side length. Since $T$ preserves both $\bar{x}$ and $a$ , the answer is $12\times 72=\boxed{1296}$ .

The value of $x_1'+x_2'+x_3'$ and $x_1'x_2' + x_2'x_3' + x_3'x_1'$ will remain the same regardless of how much you rotate $T$ . This is a consequence of François Viète's trigonomic solutions for roots to a cubic polynomial.

Viète found that, given a cubic polynomial $ax^3+bx^2+cx+d$ , if one shifts the polynomial to the right by $\frac{b}{3a}$ and defines:

$\begin{aligned} p&=\frac{3ac-b^2}{3a^2} \\ q&=\frac{2b^3-9abc+27a^2d}{27a^3} \end{aligned}$

Then the polynomial reduces to $x^3 + px + q$ . Using the trignometric identity $4 \cos^3 x - 3\cos x - \cos(3x) = 0$ , if the polynomial has 3 real roots, the roots to this second polynomial can be expressed as:

$x_i = 2\sqrt{-\frac{p}{3}}\cos\left( \frac{1}{3} \arccos \left( \frac{3q}{2p} \sqrt{\frac{-3}{p}} \right) - \frac{2 \pi i}{3} \right), \quad i=1,2,3$

Geometrically, the roots of $ax^3 + bx^2 + cx + d$ will be the x-coordinates of an equilateral triangle with a circumradius of $\displaystyle 2\sqrt{-\frac{p}{3}}$ , a centroid with x-coordinate $\displaystyle \frac{b}{3a}$ and angle of rotation of $\displaystyle \frac{1}{3}\cos^{-1}\left(\frac{3q}{2p}\sqrt{-\frac{3}{p}}\right)$ . Details of the derivation can be found here

We can define $x_1, x_2, x_3$ as the roots of the cubic $x^3 - 18x^2 + 72x + d$ , which can be seen by Vieta's Formulas, where $d$ is a value whereby the roots are real, and have $x_1, x_2, x_3$ be the x-coordinates of an equilateral triangle $T$ , whose contruction is described above. Since the circumradius and the x-coordinate of the centroid is independant of $d$ , upon changing $d$ , $T$ will simply rotate, keeping the circumradius and the x-coordinate of the centroid constant. As such, upon rotating the triangle, only $d$ changes, keeping $b = -18$ and $c=72$ constant. So $x_1', x_2', x_3'$ are the roots to the cubic $x^3 - 18x^2 + 72x + d'$ , keeping the value of $x_1'+x_2'+x_3'$ and $x_1'x_2' + x_2'x_3' + x_3'x_1'$ constant by Vieta's Formulas.

Here is a link to an interactive online visual. The value of $d$ is restrained by requiring the roots to be real, however, this does not limit the amount $T$ can rotate because the triangle is not uniquely determined by $x_i$ . By reflecting $T$ about the y-axis, this keeps keeping the x-coordinates of its vertices unchanged.