Four circles are constructed with the sides of a convex quadrilateral as the diameters. Does there exist a point inside the quadrilateral that is not inside the circles? Justify your answer?

Answer 1

4 circles of any size, put together with their diameters as sides of a quad will only take so much space.

Think about it. If a round peg is in a hole there's empty space. On the other hand if the sides are long enough, making a good-size diameter, the whole area could be covered.