Finding angles in Barycentric system How to find the angles of a triangle given the barycentric coordinates of its corners? Does it work if i take the first two components of every coordinate, and find the angles in the triangle (on the cartesian plane) determined by those coordinates?
 A: It is possible to do this by converting into Cartesian coordinates, albeit somewhat difficult, and not by the method suggested in the question.
First you have to fix the coordinates of the reference triangle, skip this step if this is already known. If all the three sides are known, you can fix $A=(0,0)$ and $AB$ on the $x$-axis, so $B=(c,0)$. Now we pick $C$ such that $AB=c$ and $AC=b$. Using the distance formula we can derive the coordinates of $C$ as, $$(X_{c},Y_{c})=(\frac{-a^2+b^2+c^2}{2c}, b^2-\frac{a^4+b^4+c^4-2a^{2}b^{2}-2a^{2}c^{2}+2b^{2}c^{2}}{2c})$$
If the reference triangle is fixed by a way different from three sides ($SAS$ or $AAS$ or $ASA$) then you will have to use the law of cosines to convert angles to sides first.
Now you can convert (homogenized) barycentric coordinates into Cartesian coordinates, by,
$$x=\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}$$
$$y=\gamma_{1}y_{1}+\gamma_{2}y_{2}+\gamma_{3}y_{3}$$
when $\gamma_{1}, \gamma_{2}, \gamma_{3}$ are the barycentric coordinates of the point we are converting, and $(x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})$ are the Cartesian coordinates of the vertices of the reference triangle.
In our case substituting gives us,
$$x=\gamma_{2}c+\gamma_{3}X_{c}$$
$$y=\gamma_{3}Y_{c}$$
Similarly you can find the Cartesian coordinates of the other two points defining the angle which is easier as you already know the required constants, find the slopes of the required two lines, and use the formula $$\text{arctan}(\pm\frac{m_{1}-m_{2}}{1+m_{1}m_{2}})$$ using the slopes as $m_{1}$ and $m_{2}$ which gives the required angle.
A: The barycentric coordinates of the corners of any triangle are always
$$[1:0:0] \qquad [0:1:0] \qquad [0:0:1]$$
So there is no angle information available. It tells you nothing about the shape of the triangle.
