Prove the equality of circle's areas

Question

Through a random point inside the circle, we draw $4$ lines with $45$ degrees between each other. Prove, that the total area of odd pieces, equals total area of even pieces.

I suppose there is a solution using integrals, but maybe there is geometric solution as well.

Answer 1

This is known as the Pizza theorem (or, at least, a special case of the pizza theorem). There is a standard proof without words, using the following image from Wikipedia: