I enjoyed Major General Panic's exposition on trisecting a line. Looks cool and is interesting!

His claim that trisecting a line is equivalent to trisecting an angle, though, is... optimistic, to put it kindly. Two non-expert refutations come to mind:

  • If trisecting an angle really is that easy, then surely he's not the first to consider using a trisected line to trisect an angle? And then the problem would long be solved.
  • If he found a way to trisect an angle, he would have disproved the proofs of those other folks who claim it's not possible. Given that mathematical proofs are usually rigorous, it's hard to envision a fundamental one like this simply falling over backwards.

I'm not a math wiz, but I think I see the fallacy in his thinking: Line AB is perpendicular to line CD, which bisects the angle ACB. So is line IJ. Simply stated, line IJ is "straight across" angle ICJ, it forms the base of an isosceles triangle. AI and JB, on the other hand, do not form isosceles triangles with the vertex at C. They're the same length as IJ, but they're not "straight across" from their respective angles; therefore, they subtend different angles! A geometry-constructing tool or careful measurement of a constructed diagram will reveal that the angles ACI and JCB are different from ICJ.

So yes, the angle is cut into three angles, but they're not equal, so the requirement of trisection is not met.