Call BE = x then AC = x
Call DE = y
Call EC = z
Triangle(ACB) is similar to triangle(DEC).
---> y / x = x / (x + z)
Right triangle ---> y = sqrt(1 - x2)
x + y + z = 2 ---> z = 2 - x - y ---> z = 2 - x - sqrt(1 - x2)
y / x = x / (x + z) ---> sqrt(1 - x2) / x = x / [ x + (2 - x - sqrt(1 - x2) ) ]
---> sqrt(1 - x2) / x = x / [ 2 - sqrt(1 - x2) ]
Cross-multiplying:
---> x2 = 2sqrt(1 - x2) - (1 - x2)
---> x2 = 2sqrt(1 - x2) - 1 + x2
---> 0 = 2sqrt(1 - x2) - 1
---> 1 = 2sqrt(1 - x2)
---> ½ = sqrt(1 - x2)
---> ¼ = 1 - x2
---> x2 = 3/4
---> x = sqrt(3)/2 [the negative answer can be ignored]
cos(ABC) = [ sqrt(3)/2 ] / 1
angle(ABC) = cos-1[ sqrt(3)/2 ]
angle(ABC) = 30o