The question
This note returns to the Pythagorean theorem through two geometric paths centred on a triangle's incircle and A-excircle. It is not presented as a new theorem: the aim is to make specific geometric relationships readable, check which steps stand, and remove claims that cannot be supported.
What the note presents
The final note keeps two complete proof variants. One develops relationships around the incenter and intercepts; the other works through the incircle, the A-excircle, and their internal homothety center. Both paths were organised after unclear, circular, or overly close formulations were removed.
How to read the result
Because the Pythagorean theorem has a long and varied proof tradition, this work does not claim an absolutely new proof or original theorem. The aim is to present the arguments clearly within classical geometry and carefully check their dependencies and possible overlap with existing ideas.