Coordinates of a vertex of a triangle?

Question

Here is the problem: There is a triangle with vertices $A,B,C$ in a cartesian coordinate system, where coordinates of points $A$ and $B$ and the angle $\alpha=\measuredangle ABC$ are given. The ratio $\frac{\overline{BC}}{\overline{AC}}$ is equal to $k$. Express coordinates of the point $C$ in terms of coordinates of points $A,B$ and $\alpha,k$.
Edit: I got an immense quadratic equation and hoped to maybe hear some suggestions about your workflow for this kind of problem.

Answer 1

Initially assume coordinates $A(1,0)$ and $B(0,0)$. (You can get any others using translations, rotations around $(0,0)$, and stretch/shrink, which all keep angles and distance ratios the same.)

Now the $\alpha$'s vertex is $(0,0)$ and the quadratic equation turns out much simpler.

EDIT: The transformations after you've calculated the simplified C:

First rotate and stretch/shrink around $(0,0)$ so that $(1,0)$ becomes $(X,Y)=(x_a-x_b,y_a-y_b)$. This is how it's done: $$(x,y) \mapsto (Xx-Yy, Yx+Xy)$$ Then translations: $$(x,y) \mapsto (x+x_b, y+y_b)$$ Easy-peasy!