**Errata in geometry books**

**Footnote 1**

**Explanation of a flaw in
Roger A. Johnson, **

Page 169, ยง264, Proof of the last Theorem.

The problem lies in the degenerate cases.

What does "It is evident intuitionally
that a minimum exists" mean? And what does "inscribed
triangle" mean?

It is indeed easy to see (by analysis) that, among all triangles
P_{1}P_{2}P_{3} such that the points P_{1}, P_{2}, P_{3} lie on the **straight lines** A_{2}A_{3}, A_{3}A_{1},
A_{1}A_{2}, there exists one
with minimum perimeter. It is also readily shown that the same
holds for triangles P_{1}P_{2}P_{3} such that the points
P_{1}, P_{2}, P_{3} lie on the **closed
segments** A_{2}A_{3},
A_{3}A_{1}, A_{1}A_{2} (a *closed segment* is a segment with its two
endpoints). But it is absolutely not clear why there is a
triangle with minimum perimeter if the points P_{1}, P_{2}, P_{3} are to lie on the **open
segments** A_{2}A_{3},
A_{3}A_{1}, A_{1}A_{2} (an *open segment* is a segment without its two
endpoints).

As a consequence of this, whether we want to or not, we have to
take into account the case when some of the points P_{1}, P_{2}, P_{3} coincide with
vertices of triangle A_{1}A_{2}A_{3}. And the proof is not
guaranteed to work in this case anymore. For instance, consider
the case when P_{1} = A_{3} and P_{2} = A_{3}, while P_{3} is the foot of the perpendicular from A_{3} to A_{1}A_{2}. The construction used in the proof, even if modified
to make sense (it is hard to speak about the lines P_{1}P_{2} and P_{1}P_{3}
making equal angles with A_{2}A_{3},
since the line P_{1}P_{2}
is not defined at all, but one can still reasonably define Q_{1}), fails to yield an
inscribed triangle with greater perimeter than P_{1}P_{2}P_{3}
in this case.

With some more work, one could characterize
such "evil" cases with the result that the triangle P_{1}P_{2}P_{3}
with minimum perimeter is either the pedal triangle of the
orthocenter of A_{1}A_{2}A_{3}, or the pedal
triangle of the vertex A_{1}, or that of the vertex A_{2}, or that of A_{3}. What remains to be shown is that it actually is the
pedal triangle of the orthocenter of A_{1}A_{2}A_{3}, and not one of the
three other options. This requires a separate demonstration!
(Besides, as we know, this holds only for acute- or right-angled
triangles A_{1}A_{2}A_{3}.)

One *can* fill in these missing steps to
obtain a complete proof, but it will not be particularly short
anymore. And it uses analysis (or intuition) at the step where
the existence of the minimal-perimeter triangle P_{1}P_{2}P_{3}
is postulated. All in all, it is a bad proof.

Errata in geometry books / footnote 1

*Darij Grinberg*

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