Why $a^2+b^2=c^2$ is named after Pythagoras? It is known by earlier generations before him such as the Chinese.

Pythagoras lived approximately $569-475$ BC. I am not aware of any evidence of Chinese mathematics having the Pythagorean theorem, or anything else for that matter, before then. It does appear that the Babylonians knew about it, though we don't know whether they proved it: indeed, as far as we know, the idea of mathematical proof was basically developed by the Greeks, starting with Thales. If not Pythagoras himself, it's likely that someone in his school was the first to prove the Pythagorean theorem.

Whether or not Pythagoras discovered his theorem is a debatable issue. But there are certain evidences that Pythagorean triples are well known before Pythagoras. The Plimpton $322$ tablet of Babylonian era, believed to be dated $1800$ BC, contains $4$ columns and $15$ rows of numbers which are regarded as Pythagorean triple. But whether they knew the relation of the number to the triangle is not known.

An Indian text Baudhayana Sulba Sutra, which dated $800$ BC, contains a list of Pythagorean triples and a statement of the Pythagorean theorem. The Apastamba Sulba Sutra (c. $600$ BC) contains a numerical proof of the general Pythagorean theorem, using an area computation.

During the Han Dynasty the Pythagorean triples were appeares in The Nine Chapters on the Mathematical Art, together with a mention of right triangles. In China, the theorem is alternately known as Shang Gao Theorem.

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