The two chords divide the circle into four regions. Let the two shaded regions have equal area $S$ each, so total shaded area $= 2S$.

Total circle area $= 2S + \text{(unshaded A)} + \text{(unshaded B)}$.

By the symmetry of the two intersecting chords and the equal shaded areas, the two unshaded regions must also be equal to each other. Therefore the area of unshaded region $A$ equals the area of unshaded region $B$, and the two quantities are equal.