Short answers, pulled from the story.
The Berlin Papyrus 6619 written around 1800 BC during Egypt's Middle Kingdom contains a problem involving two squares whose areas sum to a third square. The solution to this problem is the Pythagorean triple 6:8:10 though the text does not explicitly mention a triangle.
Euclid's Elements published around 300 BC presents the oldest extant axiomatic proof of the theorem alongside Euclid's formula for generating all primitive Pythagorean triples. English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's Elements.
Some well-known examples are 3:4:5 and 5:12:13. A Pythagorean triple has three positive integers such that their squares satisfy the equation and represents the lengths of the sides of a right triangle where all three sides have integer lengths.
According to one legend Hippasus of Metapontum was drowned at sea for making known the existence of the irrational or incommensurable. His fellows cast him overboard while he was on a voyage.
By the 1st century BC the Chinese text Zhoubi Suanjing gave a reasoning for the theorem for the 3:4:5 triangle. During the Han Dynasty which lasted from 202 BC to 220 AD Pythagorean triples appear in The Nine Chapters on the Mathematical Art.