What are the coordinates of the orthocentre of the triangle whose vertices are A(0, 0), B(4, 0) and C(1, 3)?

The orthocentre is where the altitudes meet. AB lies along the $x$-axis, so the altitude from C is the vertical line $x=1$. The side BC has slope $(3-0)/(1-4)=-1$, so the altitude from A has slope 1 and is the line $y=x$. The two meet at (1, 1), and the third altitude confirms it: AC has slope 3, so the altitude from B is $y=-(x-4)/3$, which passes through (1, 1). The trap (5/3, 1) is the CENTROID, the average of the vertices, and (2, 1) is the CIRCUMCENTRE, found from the perpendicular bisectors. As a check the three lie on one Euler line $y=1$ with the centroid dividing the join of circumcentre and orthocentre in the ratio 1 : 2. The point (1, 3) is just the vertex C.