Let $\triangle{ABC}$ be a right-triangle with $\angle{ACB}=90^\circ$. I'll be using standard notations: $a$, $b$, $c$, for the side length, $s$ for the semiperimeter, $\Delta$ for the area and r for the inradius. Then, by Poncelet theorem, $a+b=c+2r$. Squaring both sides, $a^2+2ab+b^2=c^2+4cr+4r^2$. Moreover, $\Delta=rs$ (known) implies $2ab=4r(c+r)$. Hence $a^2+b^2=c^2$.
Is this derivation of the Pythagorean theorem known?