Find triangle vertices given center and orientation

Question

If you're given the coordinates of the center of a triangle, the distance from the center to each vertex, and the angle above the horizontal and the line connecting the center and one vertex, how do you find the coordinates of each vertex of the triangle?

I want to find the coordinates of A, B, and C:

More information about the triangle that I believe is required for the math:

Answer 1

Think about how you would describe construction of the triangle corners:

1. Draw a horizontal vector of length $|OA|$: this vector has coordinates $(|OA|,0)$.
2. Rotate the vector so that it makes angle $\theta$ with the horizontal. Let's call this rotation operation $R_{\theta}$. This gives us the vector $R_\theta (|OA|, 0)$
3. Translate the vector so that it begins at the triangle center $O$: $$O + R_{\theta}(|OA|,0).$$
4. Repeat steps 1-3 for the other corners $B$ and $C$.

Finally recall that counterclockwise rotation by $\theta$ is given by the formula $$R_{\theta}(a,b) = (a\cos\theta - b\sin\theta, a\sin\theta + b\cos \theta)$$ which you can use to simplify your formulas.