In an isosceles right-angled triangle, the length of the hypotenuse is $24$. Find the area of the triangle.

Diagonals of convex quadrilateral $ABCD$ intersect at $O$. If $OA \times OD = OB \times OC$,prove that $AB \parallel CD$.

In $\Delta ABC,$ the bisectors of $\angle B$ and $\angle C$ intersect $\overline{AC}$ and $\overline{AB}$ at $D$ and $E$ respectively. If $\overline{DE} \parallel \overline{BC},$ prove that $\Delta ABC$ is an isosceles triangle.

In quadrilateral $ABCD$,a line through $A$ intersects $BD$ at $L$,$CD$ at $M$,and the extension of $BC$ at $N$. Prove that $\frac{LD^2}{LB^2} = \frac{LM}{LN}$.

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. Prove that $BD = \frac{BC \times AB}{AB + AC}$ and $DC = \frac{BC \times AC}{AB + AC}$.

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 $AC = \ldots$