# Geometric Inference of Angles in a Triangle

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 $C$.

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.

• A formula without trigonometry (kind of like a contest question?) would be possible for some values, but probably not all, as a formula would probably have the trigonometric functions in them. – S.C.B. Mar 24 '16 at 13:00
• @MXYMXY That is interesting if it is true. Some values might give rise to congruent triangles perhaps? I've tried reflecting/rotating/extending lines etc, and can't get enough information to get the extra angles. For example, extending line $ZY$ and $AB$ will create a new intersection point in the top left somewhere. But that will not add any obvious value. In fact, we could extend all the lines of the $\triangle XYZ$ and intersect them with the lines of $\triangle ABC$ and use those points to create a third triangle. But again, I can't easily infer the angles of the exterior one. – Trenin Mar 24 '16 at 13:30
• Judging by how the answer to the question appears to be irrational, I'd say an answer would require $\sin$ or $\arcsin$? – S.C.B. Mar 24 '16 at 13:32
• @MXYMXY Why would it need to be irrational? In the puzzle, the OP agrues that it is an integer and asks for an answer that can make a good argument for why it is an integer. Obviously, OP can be wrong, but I am taking it on faith that he knows what he is talking about. Maybe he can answer this question! :P – Trenin Mar 24 '16 at 13:33
• @MXYMXY I should point out that in the puzzle, the given angles were all given as integers in degrees and the answer is also an integer. – Trenin Mar 24 '16 at 13:41