<s>[INST] How to draw an ellipse if a center and 3 arbitrary points on it are given? How to draw an ellipse if a center and 3 arbitrary points on it are given?
 [/INST] The generally centered 2D ellipse equation is
$$ \begin{pmatrix} x\\y\\1 \end{pmatrix}^\top \begin{bmatrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\ 0 & 0 & -1 \end{bmatrix} \begin{pmatrix} x\\y\\1 \end{pmatrix} = 0 \\ C_{11} x^2 + C_{22} y^2 + 2 C_{12} x y - 1 = 0$$
Using the three points you can find the three coefficients $C_{11}$, $C_{22}$ and $C_{12}$. First form a 3×3 system using the above equation for the three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$.
$$\begin{bmatrix} x_1^2 & y_1^2 & 2 x_1 y_1 \\x_2^2 & y_2^2 & 2 x_2 y_2 \\ x_3^2 & y_3^2 & 2 x_3 y_3 \end{bmatrix} \begin{pmatrix} C_{11} \\ C_{22} \\ C_{12} \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$$
The above system is easily solved by inverting the 3×3 matrix.
Once the coefficients are known you can find the angle of incline of the ellipse by
$$ \theta = \frac{1}{2} \arctan \left( \frac{2 C_{12}}{C_{22}-C_{11}} \right) $$
 Note the angle is measured clockwise from horizontal 
Getting the semi-major and semi-minor axes $a$ and $b$ is a little more involved.

*

*Find $\eta = C_{11} + C_{22}$

*Find $\zeta = \frac{C_{22}-C_{11}}{\cos(2 \theta)}$

*Semi major axis $a= \sqrt{ \frac{2}{\eta - \zeta} }$

*Semi minor axis $b= \sqrt{ \frac{2}{\eta + \zeta} }$
The equation of the ellipse is now
$$ \left( \frac{(b^2-a^2) \cos^2 \theta}{a^2 b^2} + \frac{1}{b^2} \right) x^2 + \left( \frac{(a^2-b^2)\cos^2\theta}{a^2 b^2} + \frac{1}{a^2} \right) y^2 +  \frac{2 (a^2 - b^2)\sin\theta\cos\theta}{a^2 b^2} x y - 1 = 0$$
Example
Consider the points $(\sqrt{3}+1,-1)$, $(1,\sqrt{3}-1)$ and $(-1,\frac{5 \sqrt{3}}{13} + \frac{7}{13})$
Find the coefficients $$\begin{pmatrix} C_{11} \\ C_{22} \\ C_{12} \end{pmatrix} =  \begin{pmatrix} \frac{5-2\sqrt{3}}{9} \\ \frac{2 \sqrt{3}+5}{9} \\ \frac{2}{9} \end{pmatrix} $$
Find the angle $$\theta = \frac{1}{2} \arctan \left( \frac{1}{\sqrt{3}}  \right) = \frac{\pi}{12}$$
Find the axes:

*

*$\eta = \frac{10}{9}$

*$\zeta = \frac{8}{9}$

*$a^2 = \frac{2}{\frac{2}{9}}= 9$

*$b^2 = \frac{2}{2} = 1$
Find the ellipse equation
$$ \left( \frac{5}{9} - \frac{2 \sqrt{3}}{9} \right) x^2 + \left( \frac{2 \sqrt{3}}{9} + \frac{5}{9} \right)  + \frac{4}{9} x y - 1 =0 $$
Confirm results with GeoGebra:

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