Let x = the measure of angle CAE, y = the measure of angle CBD, 120 degrees - x = the measure of angle CEA, and 120 degrees - y = the measure of angle CDB.
AC/sin(120-x) = 2/sin(60) --> AC = (2/sqrt(3))(sqrt(3)cosx+sinx)
BC/sin(120-y) = 1/sin(60) --> BC = (1/sqrt(3))(sqrt(3)cosy+siny)
Now 2x+2y=120 degrees --> y=60-x -->
cosy=.5cosx+(sqrt(3)/2)sinx and siny=(sqrt(3)/2)cosx-.5sinx
So BC = (1/sqrt(3))(sqrt(3)cosx+sinx) --> AC = 2BC
Letting BC=v and AC=2v --> AB^2 = v^2 + 4v^2 - 4v^2cos60 -->
AB = v*sqrt(3) --> B is a right angle --> x=15 degrees.
Area = v^2*sqrt(3)/2 = (sqrt(3)/6)(sqrt(3)cosx+sinx)^2
Area = (sqrt(3)/6)(2+cos2x+sqrt(3)sin2x) = 1/2 + sqrt(3)/3 |