Two vertices of a triangle are $(5, -1)$ and $(-2, 3)$. If the orthocenter of the triangle is the origin $(0, 0)$,find the coordinates of the third vertex.

Find the distance of the line $4x - y = 0$ from the point $P(4, 1)$ measured along the line making an angle of $135^{\circ}$ with the positive $x$-axis.

If the orthocenter of a triangle is $(1, 1)$ and the circumcenter is $\left( \frac{3}{2}, \frac{3}{4} \right)$,find its centroid.

The centroid of the triangle $ABC$,where $A \equiv (2,3)$,$B \equiv (8,10)$,and $C \equiv (5,5)$ is

$A(1, -2), B(-2, 3), C(-1, -3)$ are the vertices of a triangle $ABC$. $L_1$ is the perpendicular drawn from $A$ to $BC$ and $L_2$ is the perpendicular bisector of $AB$. If $(l, m)$ is the point of intersection of $L_1$ and $L_2$, then $26m - 3 =$ (in $l$)

$A$ triangle has a vertex at $(1, 2)$ and the midpoints of the two sides through it are $(-1, 1)$ and $(2, 3)$. Then the centroid of this triangle is