Hot Center

If the yellow and orange areas are the same size, then $[ABC]=[ADC]$ .

Both are right triangles with identical hypotenuse, so if the areas are the same $\triangle ABC$ and $\triangle ACD$ are congruent.

$\angle BAC=\angle DAC=\dfrac{90^\circ}4=22.5^\circ$

$\overline{BC}=\overline{AB}\times\tan(22.5^\circ)=\dfrac{\tan(22.5^\circ)}{2}$

$\overline{CE}=2\overline{BC}=\tan(22.5^\circ)=\dfrac{\sqrt{2-\sqrt2}}{\sqrt{2+\sqrt2}}$

Red area $=\overline{CE}^2=\dfrac{2-\sqrt2}{2+\sqrt2}\approx\boxed{0.1716}$