If the points $A (6, 1)$,$B (8, 2)$,$C (9, 4)$,and $D (p, 3)$ are the vertices of a parallelogram,taken in order,find the value of $p$.

In $\Delta ABC$,$D$ is the midpoint of $\overline{BC}$. $A$ line passing through $G$ (the centroid) intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. If $3 AN = 2 AC$,prove that $AM = 2 MB$.

For the triangle with vertices $A (6, 7),$ $B (-2, 3)$ and $C (9, 1),$ find the coordinates of the point on $\overline{BC}$ where the bisector of $\angle A$ intersects $\overline{BC}$.

Let $A(4, 2)$,$B(6, 5)$,and $C(1, 4)$ be the vertices of $\Delta ABC$. Find the coordinates of points $Q$ and $R$ on medians $BE$ and $CF$ respectively such that $BQ : QE = 2 : 1$ and $CR : RF = 2 : 1$.

Find the value of $k$ if the points $A(2, 3)$,$B(4, k)$,and $C(6, -3)$ are collinear.

If $A$ and $B$ are $(-2, -2)$ and $(2, -4)$ respectively,find the coordinates of $P$ such that $AP = \frac{3}{7} AB$ and $P$ lies on the line segment $AB$.