A circle is divided into $5$ parts as shown in the diagram and parts are colored either red or green. Find which area is bigger.

Reflect the area $ABC$ such that $A\to A'$ and $B\leftrightarrow C$, and similarly reflect the area $CDE$ such that $C\leftrightarrow D$ and $E\to E'$. All sectional parts then have a vertex at $C$. We now have a limiting case of the Pizza Theorem ("limiting" because the center of the cutter is on the circumference). This theorem says that the white and the grey parts in the figure have the same area.

For simplicity say $r=1$.

$$Green_1 = {\pi x\over 360} - {\sin x \over 2}$$ $$Green_2={\sin (x+90) \over 2} + {\sin(180- x) \over 2} + {\pi \over 4}$$ $$Green_3 = {\pi (90-x)\over 360} - {\sin (90- x) \over 2}$$ All the $\sin $ cancels so we have $$ Green = {\pi \over 2} = Red$$

Not an answer by any stretch of the imagination, I just want to illustrate the dangers of drawing pictures that are too "nice" or too symmetrical. We could just as well have the following configuration, so there's clearly no bisector.