$\text{ABC}$ is an isosceles triangle right angled at $\text{C}$. Prove that ${\text{AB}}^2=2{\text{AC}}^2$

$\text{'}O\text{'}$ is any point in the interior of a triangle $\mathrm{ABC}.$ If $\mathrm{OD}\perp \mathrm{BC}, \mathrm{OE}\perp \mathrm{AC}$ and $\text{OF}\perp AB$ show that ${\mathrm{OA}}^2+{\mathrm{OB}}^2+{\mathrm{OC}}^2-{\mathrm{OD}}^2-{\mathrm{OE}}^2-{\mathrm{OF}}^2={\mathrm{AF}}^2+{\mathrm{BD}}^2+{\mathrm{CE}}^2$.

$\text{'}O\text{'}$ is any point in the interior of a triangle $\text{ABC}.$
If $\mathrm{OD}\perp \mathrm{BC}, \mathrm{OE}\perp \mathrm{AC}$ and $OF\perp \mathrm{AB}$ show that
${\mathrm{AF}}^2+{\mathrm{BD}}^2+{\mathrm{CE}}^2={\mathrm{AE}}^2+{\mathrm{CD}}^2+{\mathrm{BF}}^2$

In the given figure, $ABC$ is a triangle right-angled at $B. D$ and $E$ are points on $BC$ trisect it.
Prove that $8{\mathrm{AE}}^2=3{\mathrm{AC}}^2+5{\mathrm{AD}}^2$

$\mathrm{ABC}$ is an isosceles triangle right angled at $\text{B}.$ Similar triangles $\text{ACD}$ and $\text{ABE}$ are constructed on sides $\text{AC}$ and $\text{AB}.$ Find the ratio between the areas of $∆\text{ABE}$ and $∆ \text{ACD}$.