The area of a triangle having vertices $A(0,0), B(4,0)$ and $C(0,6)$ is:

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 $\overline{BC}$.

Find the circumcentre of the triangle with vertices $(0,0), (3,4), \text{and } (0,4)$.

If three vertices of a square are $(-4, 3)$,$(10, 5)$,and $(12, -9)$,then find the fourth vertex of the square.

$A(k, 2)$ and $B(3, 5)$ are the given points. If the point $P(t, t)$ divides the line segment $\overline{AB}$ in the ratio $k:1$ starting from $A$,find the value of $k$.

Find the length of the median through the vertex $(-1, 3)$ of the triangle having vertices $(-1, 3)$,$(1, -1)$,and $(5, 1)$.