The equation 3x² + 2hxy + 3y² = 0 gives two lines through the origin…

$G$ is the average of the corners. $H = 3G - 2O$, so $O$, $G$, $H$ lie on one line with $GH = 2\,OG$. $O$ is equally far from the corners. The incentre is $I = \dfrac{aA + bB + cC}{a + b + c}$, with $a$, $b$, $c$ the sides opposite $A$, $B$, $C$. The area is $\dfrac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$. For $ax^2 + 2hxy + by^2 = 0$, $\tan\theta = \dfrac{2\sqrt{h^2 - ab}}{|a + b|}$, and the slopes $m$ solve $bm^2 + 2hm + a = 0$.