The five-pointed star drawn in one unbroken line is the pentagon's set of diagonals and nothing else. Its history is long and better documented than most sacred-geometry claims.
The Pythagoreans used it as a sign of recognition. Lucian, in the second century AD, describes the figure as the triple interwoven triangle used by the school and reports that they called it health, and a scholiast on Aristophanes says the same. The Greek word hygieia has five letters, and the practice of writing one at each point is attested on later amulets, with the Latin salus doing the same job. That the Pythagoreans greeted one another with a wish for health is independently reported. The claim that the pentagram served as their badge is therefore standard and reasonably well sourced, though it rests on writers living several centuries after the school itself.
The symbol is considerably older than the Pythagoreans. Pentagram signs appear in Mesopotamian administrative texts of the fourth and third millennia BC, apparently as a sign for something like a heavenly body or a district, and the original meaning is not securely recovered.
In medieval Christian use the pentagram is a protective sign and occasionally stands for the five wounds of Christ. The fourteenth-century poem Sir Gawain and the Green Knight devotes a long passage to the pentangle on Gawain's shield, calling it Solomon's sign and reading its five points as five sets of five virtues. It is one of the fullest explanations of a symbol that anyone in the period troubled to write down.
The inversion, the idea that a point-down pentagram signifies evil, is recent and traceable. It belongs to nineteenth-century French occultism, principally to Eliphas Levi, and reached popular culture through twentieth-century channels. It is not a medieval distinction, and reading it back into earlier material is simply an error.