Why is ( BC/ DE) ^2
I am assuming you want to know by asking "why is (BC/DE)^2" (which is like asking "why is A + B?" why (BC/DE)^2 represents the ratio of the area of the smaller triangle to that of the upper triangle.
If the triangles are similar, then the proportion of any linear dimension between the two is constant. Since area is a 2 dimensional unit, the proportion between the two is that constant squared. (if b1 = kb2 and h1 = kh2, then 1/2*b1h1 = 1/2(k^2)b2h2.
Since BC/DE is the proportionallity constant between the linear dimensions of the smaller to larger triangle, (BC/DE)^2 is the proportionality constant between their areas.
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993