What is the circumcenter of the triangle with vertices $(0, 0)$,$(3, 0)$,and $(0, 4)$?

In the triangle $ABC$ with vertices $A(2, 3)$,$B(4, -1)$ and $C(1, 2)$,find the equation and length of the altitude from the vertex $A$.

Let $C(\alpha, \beta)$ be the circumcenter of the triangle formed by the lines $4x + 3y = 69$,$4y - 3x = 17$,and $x + 7y = 61$. Then $(\alpha - \beta)^2 + \alpha + \beta$ is equal to $.........$.

$O(0, 0), P(3, 4), Q(6, 0)$ are the vertices of a triangle $OPQ$. $A$ point $R$ lies inside the triangle $OPQ$ such that the areas of triangles $OPR, PQR,$ and $OQR$ are equal. Find the coordinates of $R$.

The vertices of $\Delta PQR$ are $P(2, 1)$,$Q(-2, 3)$,and $R(4, 5)$. Find the equation of the median through the vertex $R$.

If the midpoints of the sides of a triangle are $(0, 1)$,$(1, 1)$,and $(1, 0)$,find the $x$-coordinate of its incentre.