| All right, I give up.  Where's the error?

Instead of a 100 degree angle, exaggurate that to (say) 175 degrees,
and you'll see what is actually happening:

[spoiler follows]

A +-----_____
  |          -----___ D
  |                _/
  |              _/
B |_____________/
                C





	   X


The point X (the intersection of the perp. bisectors)
is actually *way* "below" ABCD,
and the segment XD actually stays outside of ABCD;
angle DCX is not DCB+BCX, but rather 360-(DCB+BCX).

In a different light, this whole thing could be viewed as a proof
that the perpendicular bisectors do not meet inside ABCD.