Problem:
Point B lies on line segment AC with AB=16 and BC=4. Points D and E lie on the same side of line AC forming equilateral triangles △ABD and △BCE. Let M be the midpoint of AE, and N be the midpoint of CD. The area of △BMN is x. Find x2.
Solution:
Place the triangles on a coordinate grid so that A is at (−16,0),B is at (0,0), and C is at (4,0). Then D is at (−8,83),E is at (2,23),M is at (−7,3), and N is at (−2,43). Because BM=MN=NB=52, △BMN is equilateral with area 43(BM)2=133=x. Thus x2=169⋅3=507.