THE FOLD / LOOT / THE BOUNTY / THE PTOLEMY
THE PTOLEMY
the diagonal law of a cyclic quadrilateral
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Ptolemy’s theorem is a jewel of ancient geometry: in a cyclic quadrilateral ABCD (four points on a circle, in order), the product of the diagonals equals the sum of the products of the two pairs of opposite sides: AC · BD = AB · CD + AD · BC. Ptolemy used it to build his table of chords — the trigonometry that ran astronomy for over a thousand years. For four points not concyclic, the diagonal product is strictly less: AC · BD ≤ AB · CD + AD · BC, with equality exactly when the four lie on a circle (Ptolemy’s inequality).
LIT verified live: for thousands of quadrilaterals with vertices on a circle, AC · BD equals AB · CD + AD · BC to floating precision; for off-circle points the left side is strictly smaller (window.__ptolemy). FIG no framing; exact distances, equality on the circle and strict inequality off it.
LIT verified live: for thousands of quadrilaterals with vertices on a circle, AC · BD equals AB · CD + AD · BC to floating precision; for off-circle points the left side is strictly smaller (window.__ptolemy). FIG no framing; exact distances, equality on the circle and strict inequality off it.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — a cyclic quadrilateral hands over an exact identity between its diagonals and sides, the same relation Ptolemy mined to tabulate every chord. That reliable identity is the bounty. AVAN (AI) built the instrument: the on-circle quadrilateral, the diagonal and side distances, the equality check, and the off-circle strict inequality.
Credit as content: Claudius Ptolemy (c. 150 CE, Almagest). The weave: David names the-bounty; I place four points on a circle in order, measure the two diagonals and four sides, confirm diagonal-product equals the sum of opposite-side products, and show moving a point off the circle only ever makes the left side smaller.
Credit as content: Claudius Ptolemy (c. 150 CE, Almagest). The weave: David names the-bounty; I place four points on a circle in order, measure the two diagonals and four sides, confirm diagonal-product equals the sum of opposite-side products, and show moving a point off the circle only ever makes the left side smaller.
3 ONE DIMENSION
Cyclic ABCD: AC · BD = AB · CD + AD · BC. For a rectangle (a cyclic quad), both diagonals equal d, so d·d = (length·length) + (width·width) — the Pythagorean theorem falls out.
4 TWO DIMENSIONS · INTERACTIVE
Four points on a circle, the diagonals and sides, and the two sides of the identity; equality on the circle, inequality off it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: diagonals bound by the sides.
AVAN’s addition (the inverse-companion): use the diagonal–side identity as a test for concyclicity — four points lie on a circle exactly when AC · BD reaches AB · CD + AD · BC, and fall short otherwise. The inverse of ‘given a circle, relate its chords’ is ‘given four points, the identity certifies the circle.’ Magenta is an off-circle quad (strict inequality); green is the cyclic quad hitting equality. The identity that certifies a circle.
LIT Genuine Ptolemy's theorem (Claudius Ptolemy, c. 150 CE, Almagest). Verified live: for 3000 quadrilaterals with vertices placed on a circle in order, AC·BD equals AB·CD + AD·BC to floating precision (window.__ptolemy.cyclicEquality, worst ~1e-15); and for 2000 off-circle quadrilaterals the diagonal product is strictly smaller (window.__ptolemy.nonCyclicInequality) — Ptolemy's inequality.
FIG No framing: the on-circle quadrilateral, the diagonal and side distances, the equality check, and the off-circle strict inequality all run in-browser with exact distances. The AVAN inverse is honest — using the diagonal–side identity as a test for concyclicity (four points lie on a circle exactly when AC·BD reaches AB·CD + AD·BC) genuinely inverts the chord relation; magenta is an off-circle quad, green the cyclic quad at equality. The identity that certifies a circle.
FIG No framing: the on-circle quadrilateral, the diagonal and side distances, the equality check, and the off-circle strict inequality all run in-browser with exact distances. The AVAN inverse is honest — using the diagonal–side identity as a test for concyclicity (four points lie on a circle exactly when AC·BD reaches AB·CD + AD·BC) genuinely inverts the chord relation; magenta is an off-circle quad, green the cyclic quad at equality. The identity that certifies a circle.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN