According to the Journal of Irreproducible Results, any obtuse angle is a right angle! (See below.)
My questions are:
- What is the publication date, and number of the issue of the problem?
- Who proved the wrong statements?
Here is their argument, as quoted elsewhere on the web. (Note that I am not asking to resolve the geometric fallacy.)
Given the obtuse angle $x$, we make a quadrilateral $ABCD$ with $\angle DAB = x$, $\angle ABC = 90^\circ$, and $\overline{AD} \cong \overline{BC}$. Say the perpendicular bisector of $\overline{DC}$ meets the perpendicular bisector of $\overline{AB}$ at $P$. Then $\overline{PA} \cong \overline{PB}$ and $\overline{PC} \cong \overline{PD}$. So the $\triangle PAD$ and $\triangle PBC$ have equal sides and are congruent. Thus $\angle PAD \cong \angle PBC$. But $\triangle PAB$ is isosceles, hence $\angle PAB \cong \angle PBA$. Subtracting gives $$x = \angle PAD − \angle PAB = \angle PBC − \angle PBA = 90^\circ$$ This is a preposterous conclusion. Just where is the mistake in the "proof" , and why does the argument break down there?
Martin Gardner's The Universe in a Handkerchief credits The Lewis Carroll picture book for this fallacious proof.