Thābit ibn Qurra's Generalization of the Pythagorean Theorem

Thābit ibn Qurra's theorem generalizes the Pythagorean theorem to any triangle ABC. Draw lines from A that meet side BC at points B′ and C′, making angles equal to angle A. Then AB² + AC² = BC · (BB′ + CC′). When angle A is a right angle, B′ and C′ coincide and the formula becomes the Pythagorean theorem formula, a² + b² = c².

Thābit ibn Qurra (c. 826/836–901) worked in Baghdad at the height of the Abbasid translation movement. He stated this generalization in a short treatise that also gave his own proofs of the ordinary theorem — Euclid's Proposition I.47 — and the result still gives a neat route to the law of cosines.

Who was Thābit ibn Qurra?

Thābit was born in Harran, in upper Mesopotamia (today in southeastern Turkey), in 826 or 836 — the sources disagree. He belonged to the Sabians of Harran, a community that kept an old star-centered religion and a long tradition of Greek learning. His first language was Syriac, and he also knew Greek and Arabic, which made him an ideal translator.

According to the standard accounts, Muḥammad ibn Mūsā ibn Shākir, one of three wealthy brothers known as the Banū Mūsā, met him in Harran, was impressed by his languages and brought him to Baghdad. There Thābit studied mathematics with the brothers and joined their circle of translators. Later he became court astronomer to the caliph al-Muʿtaḍid (reigned 892–902). He died in Baghdad in 901.

Baghdad and the translation movement

From the late eighth century to the tenth, scholars in Baghdad translated a huge body of Greek science and philosophy into Arabic, often by way of Syriac, paid for by caliphs and private patrons such as the Banū Mūsā. The period is often associated with the "House of Wisdom" (Bayt al-Ḥikma); historians still debate whether that was a research academy or mainly a palace library, but there is no doubt about the scale of the translation work.

Thābit was one of its leading figures. He revised an earlier Arabic translation of Euclid's Elements; the resulting text, often called the Isḥāq–Thābit version, became one of the standard Arabic Euclids. He also translated or revised works of Archimedes, Apollonius and Ptolemy. His own research ranged from statics (the theory of the balance) to number theory, where he found a rule for producing amicable numbers — pairs like 220 and 284, each equal to the sum of the other's proper divisors.

The theorem for any triangle

Start with any triangle ABC and focus on the angle at A; call it α. From A, draw two lines to side BC:

• one meeting BC at B′, so that the angle AB′B equals α;
• one meeting BC at C′, so that the angle AC′C equals α.

Thābit's theorem says that the squares on the two sides meeting at A add up to a rectangle built on the third side:

Obtuse, right and acute: one statement, three cases

The beauty of Thābit's version is that one formula covers every shape of triangle. What changes is how the segments BB′ and CC′ sit along BC:

• Angle A obtuse: the two segments leave a gap, so BB′ + CC′ < BC and AB² + AC² < BC².
• Angle A right: both lines become the altitude from A, so B′ = C′ and BB′ + CC′ = BC. The formula is the Pythagorean theorem, AB² + AC² = BC².
• Angle A acute: the segments overlap, so BB′ + CC′ > BC and AB² + AC² > BC².

This is the same test that tells you whether a triangle is acute, obtuse or right from its side lengths, and the right-angle case is the converse of the Pythagorean theorem — now with a picture of why it works. For an acute example, take AB = 6, AC = 7, BC = 8: then BB′ = 36 over 8 = 4.5 and CC′ = 49 over 8 = 6.125, which overlap by 2.625, and 8 × 10.625 = 85 = 36 + 49.

Why it works: two pairs of similar triangles

The proof needs nothing more than similar triangles. Write α for angle BAC.

1. Compare triangle ABB′ with triangle CBA. They share the angle at B, and by construction the angle at B′ equals α, the angle at A in the big triangle. Two equal angles make the triangles similar, with B′ matching A, B matching B and A matching C.
2. Read off a proportion. Matching sides give BB′ over AB = AB over BC, so AB² = BC · BB′.
3. Do the same on the other side. Triangle ACC′ shares the angle at C with triangle BCA and has angle α at C′, so CC′ over AC = AC over BC and AC² = BC · CC′.
4. Add the two equations. AB² + AC² = BC · BB′ + BC · CC′ = BC · (BB′ + CC′).

When α = 90°, B′ and C′ are both the foot of the altitude, and steps 2 and 3 are the two halves of the geometric mean (leg) theorem. In other words, the standard similar-triangles proof of the Pythagorean theorem is Thābit's proof in its special case.

The link to the law of cosines

The difference between BC and BB′ + CC′ is not just "a gap" or "an overlap"; it has an exact size. Using BB′ = AB²/BC and CC′ = AC²/BC:

By the law of cosines, BC² − AB² − AC² = −2 · AB · ACcosα. So the gap between B′ and C′ is −2 · AB · ACcosα ÷ BC: positive when cosα < 0 (obtuse), zero at 90°, negative — an overlap — when α is acute. For the 5-6-9 triangle, cosα = −1 over 3 and the gap is 20 over 9 ≈ 2.22, matching the figure.

Euclid had already covered the obtuse and acute cases separately in Propositions II.12 and II.13 of the Elements, using rectangles rather than cosines. Thābit's formula packs both, and I.47, into a single statement. Another well-known extension, Pappus's area theorem, replaces the squares with parallelograms; see the theorem with other shapes.

Thābit's own proofs of the right-angle theorem

Thābit's generalization comes from a short treatise that historian Aydın Sayılı described in 1960. As Sayılı and later writers report it, the treatise is written as a letter to a friend. It begins with the so-called proof of Socrates — the special case of the isosceles right triangle, from the square-doubling lesson in Plato's Meno — and then gives general proofs of the theorem before extending it to all triangles.

A dissection proof very close to the one animated below appears in a manuscript of Thābit's preserved in Istanbul, as described by R. Shloming in 1970. Put the squares on the two legs, a² and b², side by side. Cut off two copies of the right triangle, then swing each one through 90° about a corner. The pieces now form the square on the hypotenuse.

The same arrangement of pieces turns up when the squares are repeated across the plane in the Pythagorean tiling, a floor pattern that proves the theorem. Compare it with the rearrangement proof and Euclid's own proof, which Thābit knew well from the Elements he had revised.

After Thābit: al-Nayrīzī and Naṣīr al-Dīn al-Ṭūsī

Work on Euclid, and on I.47 in particular, continued for centuries in the Islamic world.

al-Nayrīzī (c. 865 – c. 922)

A Persian mathematician and astronomer from Nayriz, a younger contemporary of Thābit who also wrote a work for the caliph al-Muʿtaḍid. His commentary on the Elements (based on al-Ḥajjāj's Arabic translation) survives, preserves extracts from the lost Greek commentaries of Heron and Simplicius, and was translated into Latin by Gerard of Cremona in the twelfth century. Roger Nelsen credits him with a proof of the theorem based on the Pythagorean tiling.

Naṣīr al-Dīn al-Ṭūsī (1201–1274)

Born in Tus in Khorasan, he directed the observatory at Maragha (begun in 1259) and wrote a widely studied recension of Euclid's Elements. An Arabic Euclid printed in Rome in 1594 was long credited to him, but many historians now attribute it to a later author, often called "pseudo-Ṭūsī". His Treatise on the Quadrilateral presents trigonometry as a subject in its own right, including the law of sines for plane triangles.

Arabic versions of the Elements were also the route by which much of Euclid reached Latin Europe: Adelard of Bath and Gerard of Cremona translated from Arabic in the twelfth century. When medieval European scholars first studied I.47, they were often reading it through the work of Baghdad translators like Thābit. For the wider story, see the history of the Pythagorean theorem.