The easiest way I can think of is to use vector math to find BC (BA+AC=BC) and AD (AD = DB-BA) and get the intersection E from them. You can find the analogue to point C on the circle around D (call it G) to construct a line BG and F would be the intersection of BG and AD.

I wrote an article that details a method that simply uses the cross product to determine the intersection of 2 lines. The lines need to be in an implicit form (ax+by+c=0) and the resulting point is in homogeneous coordinates (x*w,y*w,w), but those transformations are very simple. This method also doesn't yield any infinities if the lines are parallel, and no trigonometric functions are used.