In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x-3$,$MB = 2x-7$,$AN = x+3$ and $NC = 2x+3$,find the value of $x$.

In $\Delta ABC$,$A-D-B$,$A-E-C$ and $BD = CE$. If $m\angle B = m\angle C$,prove that $\overline{DE} \parallel \overline{BC}$.

The length of a diagonal of a square is $6$. Then, the length of its sides is $\ldots \ldots \ldots$ (in $\sqrt{2}$)

In an equilateral triangle,prove that three times the square of one side is equal to four times the square of one of its altitudes.

In $\Delta ABC$,medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$. $A$ line through $G$ and parallel to $\overline{DE}$ intersects $\overline{AC}$ at $K$. If $EK = 1.8$,then $AC = \ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $AB = 6$ and $AC = 10$,find the area of $\Delta ABC$.