The incentre of the triangle formed by the vertices $(0, 0)$,$(5, 12)$,and $(16, 12)$ is:

If $(a, b - c)$,$(b, c - a)$,and $(c, a - b)$ are the vertices of a triangle,then where does the centroid of the triangle lie?

Let $O(0,0), P(3,4), Q(6,0)$ be the vertices of the triangle $OPQ$. The point $R$ inside the triangle $OPQ$ is such that the triangles $OPR, PQR, OQR$ are of equal area. The coordinates of $R$ are

Assertion: If $(0, 3), (1, 1)$ and $(-1, 2)$ are the midpoints of the sides of a triangle,then the centroid of the original triangle is $(0, 2)$.
Reason: The centroid of a triangle and the centroid of the triangle formed by joining the midpoints of the sides of the original triangle are the same.

Consider a $\triangle PQR$ in which the relation $QR^2 + PR^2 = 5PQ^2$ holds. Let $G$ be the point of intersection of medians $PM$ and $QN$. Then,$\angle QGM$ is always

The distance of the origin from the centroid of the triangle whose two sides have the equations $x - 2y + 1 = 0$ and $2x - y - 1 = 0$ and whose orthocenter is $\left(\frac{7}{3}, \frac{7}{3}\right)$ is