I borrowed this from http://fivetriangles.blogspot.com/

184. *** Overlapping sectors

In the diagram is rectangle ABCD with height 10 cm. An arc with centre at point B is drawn from point A to side BC. An arc with centre at point C is drawn from point D to side BC. Given that the shaded (coloured) regions ⓐ and ⓑ have equal area, determine the length of BC.

The extension is “and where do the quarter circles cross?”

It could be a two line calculation !