Regular polygons and Pythagoras Let $L_n:$ the side length of a regular $n$-polygon inscribed in a  unit
fixed circle.
We have an interesting relationship:
$L_6^2+L_6^2=L_4^2$
$L_6^2+L_4^2=L_3^2$
$L_{10}^2+L_6^2=L_5^2$

There are more solutions: $L_m^2+L_n^2=L_p^2$ ?
This will not answer the question as phrase, but:


The last one you list is Proposition 10 of Book XIII of Euclid's Elements.  It was used by Ptolemy in the second century for constructing a trigonometric table.
The circle need not be of unit radius.  You could have just said "inscribed in a circle".
This will not answer the question as phrase, but:


The last one you list is Proposition 10 of Book XIII of Euclid's Elements.  It was used by Ptolemy in the second century for constructing a trigonometric table.
The circle need not be of unit radius.  You could have just said "inscribed in a circle".