Why is the circumcentre always taken as the origin?

Question

For instance, I have a question which asks about locus of orthocentre of variable triangle with vertices $A(3,4), B(5 \cos\theta, 5 \sin\theta)$ and $C (5 \sin\theta,-5 \cos\theta)$. In the solution the origin $(0,0)$ is taken as the circumcentre. I don't really understand this. Why is this so?

Answer 1

1) You can set the circumcenter to be $(0,0)$ - if you placed your triangle in some other position $(x_0,y_0)$, you'd be able to simply move your triangle by $(-x_0,-y_0)$; it doesn't change the triangle itself, and moves the circumcenter to $(0,0)$.

2) It's a good idea to do so - it often makes calculations a lot nicer. For example, on this problem, you may utilize the fact that the centroid (easy to calculate regardless of where the points are) is $1/3$ of the way from the circumcenter to the orthocenter, which becomes very nice when you already know the circumcenter!