$ABC$ is an isosceles triangle right angle at $C$. Prove that $A{B}^2=2A{C}^2$.

In the figure, the perpendicular from $A$ on side $BC$ of a $△ABC$, intersects $BC$ at $D$ such that $DB=3CD$. Prove that $2A{B}^2=2A{C}^2+B{C}^2$.

$ABC$ is a right triangle right angled at $B$. Let $D$ and $E$ be any points on $AB$ and $BC$ respectively. Prove that ${\mathrm{AE}}^2+{\mathrm{CD}}^2={\mathrm{AC}}^2+{\mathrm{DE}}^2$.

$ABC$ is an isosceles triangle right-angled at $B$. Equilateral triangles $ACD$ and $ABE$ are constructed on sides $AC$ and $AB$. Find the ratio between the areas of $∆ABE$ and $∆ACD$.

$△ABC~△DEF$. $BC=3 \mathrm{cm}, EF=4 \mathrm{cm}$ and area of $△ABC=54 {\mathrm{cm}}^2$. Determine the area $△DEF$.