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What relationship did Archimedes discover between a parabolic segment and a triangle with the same base and height?

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Portrait of Archimedes by Giuseppe Nogari shows an elderly man with white hair and beard, holding a compass, gazing downward, set against a dark background. — Archimedes Archimedes, oil on canvas by Giuseppe Nogari, 18th century; in the Pushkin Fine Arts Museum, Moscow. Archimedes was an ancient Greek mathematician and inventor.

Archimedes discovered that the area of any segment of a parabola is exactly $4 / 3$ of the area of the triangle that shares the same base and height as that segment.12

He demonstrated this result in his treatise Quadrature of the Parabola, initially using a "mechanical" method of discovery and subsequently confirming it through conventional geometric proofs. His calculation effectively utilized the method of exhaustion and involved the summation of an infinite geometric series ( $1 + 1 / 4 + 1 / 16 + 1 / 64 + \dots = 4 / 3$ ) to determine the area.12