In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. If $\frac{MB}{MD} = \frac{5}{3}$ and $CD = 12$,find $AB$.

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 rhombus $ABCD$,$AC = 18$ and $BD = 24$. Find the perimeter of rhombus $ABCD$.

In $\Delta ABC$,$\overline{AD}$ and $\overline{BE}$ are medians. Line $m$ through $D$ and parallel to $\overline{BE}$ intersects $\overline{AC}$ at $K$. If $EK = 2.5$,then $AC = \ldots$

In $\Delta ABC$,$m\angle B = 90^{\circ}$,$N \in \overline{AB}$ and $M \in \overline{BC}$. Prove that $AM^{2} + CN^{2} = AC^{2} + MN^{2}$.

In $\Delta ABC$,$m \angle A + m \angle C = m \angle B$ and $\overline{BM}$ is an altitude to $\overline{AC}$. If $AM = 16$ and $CM = 9$,find $BM$,$AB$,and $BC$.