For $\Delta ABC$ and $\Delta XYZ$,$\frac{AB}{XZ} = \frac{BC}{XY} = \frac{AC}{YZ}$. Then,the correspondence $ABC \leftrightarrow \ldots \ldots$ between them is a similarity.

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 a rectangle $ABCD$,if $AB^{2} + BC^{2} = 64$,then find the length of the diagonal $AC$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$,$\overline{BM}$ is an altitude to the hypotenuse $AC$,and $AM < CM$. If $BM = 6$ and $AC = 13$,find $AB$.

In rhombus $ABCD$,$AC = 24$ and $BD = 70$. Find the perimeter of rhombus $ABCD$.

In $\Delta ABC$ and $\Delta PQR$,if $\frac{AB}{PQ} = \frac{BC}{PR} = \frac{CA}{QR}$,then the correspondence $ABC \leftrightarrow \dots$ is a similarity.