The centres of a triangle in code

A computer finds the four centres of any triangle in a moment. In this lab you write the functions for the area, the centroid, the circumcentre and the orthocentre, and the checker tries them on the triangle of Example 1 and on random triangles.

Your tasks.

1. In area, return the area from the corner formula.
2. In centroid, return the average of the three corners, as a pair.
3. In circumcentre, solve the two perpendicular-bisector equations $2(x_2 - x_1)x + 2(y_2 - y_1)y = x_2^2 + y_2^2 - x_1^2 - y_1^2$ for the pairs $(A, B)$ and $(A, C)$.
4. In orthocentre, use $H = 3G - 2O$.