NOTE: This question is related to a puzzle. If you are trying to solve that puzzle and don't want to be spoiled, then don't read any answers here, assuming there are any. In trying to solve it, I came up with this generic question which should have an answer if that puzzle is valid, but is eluding me.
You are given a triangle $\triangle ABC$ and two points on different line segments of the triangle $X$ and $Y$. You know all the angles of $\triangle ABC$, and you know the angles at points $X$ and $Y$ involving $A$, $B$ and $C$.
Here is what you start with.
You are given: $$\angle ABC, \angle BCA$$ $$\angle XCA, \angle YAC$$
You can then infer the remaining angles.
Call the intersection inside the triangle the point $O$. Now draw a line from $B$ through $O$, and call the point $Z$ where this line intersects with the $AC$ line segment.
Lastly, create a new triangle $\triangle XYZ$. It looks like this: Obviously, this triangle is uniquely defined and it should be possible to compute its angles as well as all the angles involving point $O$.
Is there a way geometrically and/or with a system of equations to get values for all these angles as functions of the given angles? Ideally, without using trigonometric functions in the final answers. Specifically, the puzzle in question is asking for $\angle XYO$, and that value could be easily used to infer the others.