I have read most of the literature, including your various answers, 
that "insist" that an angle cannot be trisected. Nevertheless, I 
believe I have the solution to this age-old problem. It seems to work 
on any angle except a right angle. It does not involve any of the 
problems I've seen described in the literature and is in strict 
compliance with the use of only a straightedge and a compass. I 
developed the solution myself.
I will provide my construction without benefit of a diagram. If you follow the steps 
carefully, you will be able to construct the diagram without much difficulty. Although this works on any angle except a straight angle, I suggest for STEP 1 you use an angle of about 70 to 90 degrees and for STEP 2 a circle with radius of about 2 inches:

STEP 1: Draw angle BAC.

STEP 2: From point A, draw circle to intersect line BA at point D 
 and line CA at point E.

STEP 3: Extend line BDA until it intersects circle at point F.

STEP 4: Extend line CEA until it intersects circle at point G.

STEP 5: Bisect arc DE at point H.

STEP 6: Draw lines GD and FH until they intersect at point I.

STEP 7: Draw lines FE and GH until they intersect at point J.

STEP 8: Draw lines AI and AJ.

STEP 9: Angle BAI = angle IAJ = angle JAC!?!(by measuring with a compass)
Do you find any hole or flaw?