Geometry problem, borrowed and extended. Try it !

I borrowed this from http://fivetriangles.blogspot.com/
184. *** Overlapping sectors
overlappingsectorsspic
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 ā“ and ā“‘ 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 !

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