Thābit ibn Qurra's Theorem

Thābit ibn Qurra's theorem generalizes the Pythagorean theorem to an arbitrary triangle . Choose points and on the line so that . The resulting similar triangles give

(1)

and

(2)

Adding the two identities yields

(3)

When is a right angle, the construction reduces to the usual Pythagorean theorem. Thabit ibn Qurra gave this result in a ninth-century treatise on the proof attributed to Socrates (Sayili 1960, Maor 2007). Three manuscript witnesses of the treatise are cataloged in Cairo, Damascus, and the Süleymaniye Manuscript Library in Istanbul (Islamic Scientific Manuscripts Initiative).