<s>[INST] Triangles of the form $a^n+b^n=c^n$ 
Let $\Delta$ be a triangle with (real) sides length $a,b,c>0$. From this question, we know that $$a+b=c \qquad \iff \qquad \Delta \text{ is a straight line}$$ Now, we know from the Pythagorean theorem that 
  $$ a^2+b^2=c^2 \qquad \iff \qquad \Delta \text{ is a right triangle}$$
  I'm wondering if it is possible, for a fixed $n>2$, to have a geometric characterization of the triangles which satisfy $a^n+b^n=c^n$, that is:
  $$ a^n+b^n=c^n \qquad \iff \qquad \Delta \text{ is ?}.$$

Notes: For $n>1$, and any $a,b>0$ there exists always a non-degenerated triangle $\Delta$ with $c= (a^n+b^n)^{1/n}$. It seems that for the case $n=3$, things are already more complicated. All the following triangles satisfy 
$$a^3+b^3=c^3.$$

 [/INST] If $x = \frac{a}{c}$, $y = \frac{b}{c}$ (note: both are in $(0, 1)$)
then for $n \gt 2$,
$x + y \gt x^2 + y^2 > x^n + y^n = 1$
Thus $a + b \gt c$ and $a^2 + b^2 \gt c^2$ and so $a,b,c$ form the sides of a triangle and the triangle must be acute.
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