TITLE: Trisecting an angle
QUESTION [7 upvotes]: In this Numberphile video it is stated that trisecting an angle is not possible with only a compass and a straight edge. Here's a way I came up with:

Let the top line be A and bottom line be B, and the point intersecting P.
  1. Use the compass and pick any length. Draw a point M n units from P on A. Use the compass again to draw a point N n units from P on B.
  2. Connect M and N.
  3. Since we know how to trisect a line, we can trisect it and get 3 equal distance line segments with 2 points in between.
  4. Connect the 2 points to the point P.
  5. Done.

This seems to work for all angles that are not 0 or 180 degrees.
Given that it is proven that it's not possible to trisect an angle with only compass and a ruler without marks, something must have been wrong in my steps but I can't see any. Where is my mistake?

REPLY [9 votes]: Let $X$ and $Y$ be the two points you added to $\overline{MN}$, as shown. 

Consider $\triangle PMY$. Assuming your trisection construction to be valid, $\overline{PX}$ must bisect $\angle MPY$. By the Angle Bisector Theorem, we have
$$\frac{|\overline{PM}|}{|\overline{PY}|} = \frac{|\overline{XM}|}{|\overline{XY}|} = 1$$
so that $\overline{PM}\cong\overline{PY}$. By the same logic, $\overline{PN}\cong\overline{PX}$. Thus, $M$, $N$, $X$, $Y$ are all equidistant from $P$.
See the problem?