Distance Formula and Section Formula in 3D

The incentre of the triangle with vertices $\left(1\text{,}\sqrt{3}\right)$, $(0, 0)$ and $(2, 0)$ is

The points $\mathrm{A}(2,3,4),\mathrm{B}(-1,2,-3)$ and $\mathrm{C}(-4,1,-10)$ are collinear.

If $D, E, F$ are midpoints of sides $BC, CA$ and $AB$ respectively of triangle $ABC$, then prove that $\overline{DG} + \overline{EG}+\overline{FG} = \overline{0}$.

If $ABC$ is a triangle whose orthocenter is $P$ and the circumcenter is $Q$, then prove that $\overline{PA}+\overline{PC}+\overline{PB}=\overline{PQ}$.

If $A, B, C, D$ are four non-collinear points in the plane such that $\overline{AD} + \overline{BD} + \overline{CD} = \overline{0}$, then prove that the point $D$ is the centroid of the triangle $ABC$.

If $G_1$ and $G_2$ are centroids of the triangles $ABC$ and $PQR$ respectively, then prove that $\overline{AP}+\overline{BQ}+\overline{CR}=3{\overline{G}}_1\overline{G_2}$.

Verify whether the four points $\left(5, -1, 1\right), \left(-1, -3, 4\right),$ $\left(7, -4, 7\right)$ and $\left(1, -6, 10\right)$ are the vertices of rhombus or not.

Show that the points $\left(0, 7, 10\right), \left(-1, 6, 6\right)$ and $\left(-4, 9, 6\right)$ form a right angled triangle.

Show that points $\left(2, -1, -1\right), \left(4, -3, 0\right)$ and $\left(0, 1, -2\right)$ are collinear.

Find the values of $\lambda$ for which the points $(6, -1, 2),(8, -7, \lambda )$ and $\left(5, 2, 4\right)$ are collinear.

The locus of a point $P$ such that $PA+PB=4$ where $A\left(2, 3, 4\right)$ and $B\left(-2, 3, 4\right)$ is

The equation of the locus of a point $P\left(x, y, z\right)$ such that it's distance from the $x$-axis is equal to its distance from the plane $x+z=1$ is

Find the position vector of a point $R$  which divides the line joining two points $P$ and $Q$  whose position vectors are $P(\hat{i}+2\hat{j}-\hat{k})$ and $Q(-\hat{i}+\hat{j}+\hat{k})$ respectively, in the ratio $2 : 1$   externally

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $P(\hat{i}+2\hat{j}-\hat{k})$ and $Q(-\hat{i}+\hat{j}+\hat{k})$ respectively, in the ratio $2 : 1$  internally

$ABCD$ is a parallelogram. $L$ is a point on $BC$ which divides $BC$ in the ratio$1:2$ . $AL$ intersects $BD$ at $P.M$ is a point on $DC$ which divides $DC$  in the ratio $1:2$ and $AM$ intersects $BD$ in $Q$.

Point $Q$ divides $DB$ in the ratio

$ABCD$ is a parallelogram. $L$ is a point on $BC$ which divides $BC$ in the ratio$1:2$ . $AL$ intersects $BD$ at $P.M$ is a point on $DC$ which divides $DC$  in the ratio $1:2$ and $AM$ intersects $BD$ in $Q$.

 Point $P$ divides $AL$ in the ratio

The coordinates of points of trisection of the line joining the points $A(x_1,y_1,z_1) \& B(x_2,y_2,z_2)$ are the points which divide $A \& B$ in the ratio $1:3 \& 3:1$ externally.

Find the coordinates of points of trisection of line joining the points $(2,3,4) \& (1,2,7)$

if the points $(1,1, \lambda )$ and $(-3,0,1)$ are equidistant from the plane $\overrightarrow{r}.\left(3\hat{i}+4\hat{j}-12\hat{k}\right)+13=0$, find the value of $\lambda$.

If the distance of the point $\hat{i}+2\hat{j}-\hat{k}$ from the plane $\overrightarrow{r}·\left(\hat{i}-2\hat{j}+4\hat{k}\right)=10$ is $\frac{17}{\sqrt{k}}$, then the find the value of $k$.