The question
This note returns to the Pythagorean theorem through two geometric paths centered 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 organized 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. It is a careful proof organization and expression attempt within a classical geometry setting, with the dependency structure and possible overlap with existing ideas checked as closely as possible.