Incircle–Excircle Pythagorean Proof Project

A mathematical exploration around the Pythagorean theorem

This project begins from the Pythagorean theorem and tries to reorganize two proof paths within a classical geometry frame. The directions that remained in the end mainly revolve around the incircle and the A-excircle of a triangle. It is not meant to claim a completely new theorem, but to place a familiar problem back into specific geometric relationships and observe which ideas can stand, and which expressions need to be removed or tightened.

What the exploration did

During the process, I tried different geometric entry points and gradually removed parts that were unclear, likely to become circular, or too close to existing ideas. In the end, the project kept two relatively complete proof paths: one around the incenter and intercept relationships, and another around the relationship between the incircle and the A-excircle.

The full proof details are not unfolded here. They were later organized into a public mathematical note.

What remained in the end

After several rounds of organization, searching, and comparison, the project finally kept two relatively complete geometric proof paths. Both are built around the incircle and the A-excircle of a triangle.

In this process, I also repeatedly checked dependency relationships inside the proofs, possible circular reasoning, and the relationship between these paths and existing geometric ideas. Because the Pythagorean theorem already has a very long proof tradition, we cannot determine that the result is absolutely a new proof or an original theorem. I therefore place it as a proof organization and expression attempt within a classical geometry frame.

For me, this is also one of the most important parts of the project: not only keeping the result of a proof, but also the process and the experience gained from it.