If $A(0,4,0)$,$B(0,0,3)$,and $C(0,4,3)$ are the vertices of $\Delta ABC$,then its incentre is

The triangle formed by the points $(0, 7, 10), (-1, 6, 6), (-4, 9, 6)$ is

If the origin is the centroid of the triangle $PQR$ with vertices $P(2a, 2, 6)$,$Q(-4, 3b, -10)$,and $R(8, 14, 2c)$,then find the values of $a, b$,and $c$.

Let $A(x, y, z)$ be a point in the $xy$-plane,which is equidistant from three points $P(0, 3, 2)$,$Q(2, 0, 3)$,and $R(0, 0, 1)$. Let $B = (1, 4, -1)$ and $C = (2, 0, -2)$. Then among the statements $(S1) :$ $\triangle ABC$ is an isosceles right-angled triangle and $(S2) :$ the area of $\triangle ABC$ is $\frac{9 \sqrt{2}}{2}$.

If $A \equiv (5, 1, p)$,$B \equiv (1, q, p)$,and $C \equiv (1, -2, 3)$ are the vertices of a triangle and $G \equiv (r, -\frac{4}{3}, \frac{1}{3})$ is its centroid,then the values of $p, q, r$ are respectively:

$G(1,0,1)$ is the centroid of the triangle $ABC$. If $A=(1,-4,2)$ and $B=(3,1,0)$,then $AG^2+CG^2=$