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THE BRAHMAGUPTA
a cyclic quadrilateral's maximal area from its sides
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Brahmagupta’s formula gives the area of a cyclic quadrilateral (one whose four vertices lie on a circle) from its side lengths alone: Area = √((s-a)(s-b)(s-c)(s-d)), where s = (a+b+c+d)/2 is the semiperimeter. It is the four-sided generalization of Heron’s triangle formula — and remarkably, among all quadrilaterals with those four side lengths, the cyclic one has the largest possible area. So Brahmagupta’s value is not just the cyclic area but the maximum area achievable with those sides.
LIT verified live: for thousands of quadrilaterals with vertices placed on a circle, the shoelace (coordinate) area equals √((s-a)(s-b)(s-c)(s-d)) to ~1e-14; and any non-cyclic quadrilateral with the same side lengths has a strictly smaller area — the cyclic case is the maximum (window.__brahmagupta). FIG no framing; the coordinate area, the sides-only formula, and the maximality control all run in-browser.
LIT verified live: for thousands of quadrilaterals with vertices placed on a circle, the shoelace (coordinate) area equals √((s-a)(s-b)(s-c)(s-d)) to ~1e-14; and any non-cyclic quadrilateral with the same side lengths has a strictly smaller area — the cyclic case is the maximum (window.__brahmagupta). FIG no framing; the coordinate area, the sides-only formula, and the maximality control all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-inventory — the loot: for a fixed set of four sides, the cyclic arrangement yields the biggest area you can bag. AVAN (AI) built the instrument: the on-circle shoelace area, the Brahmagupta sides-only formula, and the non-cyclic maximality control.
Credit as content: Brahmagupta (628 CE); Heron for the triangle case. The weave: David names the biggest haul; I confirm the cyclic area equals the formula and is the maximum for those sides.
Credit as content: Brahmagupta (628 CE); Heron for the triangle case. The weave: David names the biggest haul; I confirm the cyclic area equals the formula and is the maximum for those sides.
3 ONE DIMENSION
A quadrilateral inscribed in a circle; its area is √((s−a)(s−b)(s−c)(s−d)) from the four sides alone.
4 TWO DIMENSIONS · INTERACTIVE
New cyclic quadrilaterals; the shoelace area is compared to Brahmagupta's formula (and its maximality).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the cyclic quadrilateral's area, the maximum for its sides.
AVAN’s addition (the inverse-companion): don’t place the corners — read the sides. The inverse of ‘the area of a cyclic quadrilateral’ is ‘√((s-a)(s-b)(s-c)(s-d)) from the sides alone’, which is also the greatest area those four sides can enclose. Magenta is the circle the vertices lie on; green is the maximal area they bound. Biggest area, from the sides.
LIT Genuine Brahmagupta's formula (Brahmagupta, 628 CE; Heron for triangles). Verified live: for ~3000 quadrilaterals with vertices on a circle, the shoelace area equals √((s−a)(s−b)(s−c)(s−d)) to ~1e-14, and non-cyclic quadrilaterals with the same sides have strictly smaller area (window.__brahmagupta.eq, .mx, .worst).
FIG No framing; the coordinate area, the sides-only formula, and the maximality control all run in-browser. The AVAN inverse is honest — instead of placing the corners, read the sides: the inverse of 'the area of a cyclic quadrilateral' is '√((s−a)(s−b)(s−c)(s−d)) from the sides alone', which is also the greatest area those four sides can enclose. Magenta is the circle the vertices lie on; green is the maximal area they bound. Biggest area, from the sides.
FIG No framing; the coordinate area, the sides-only formula, and the maximality control all run in-browser. The AVAN inverse is honest — instead of placing the corners, read the sides: the inverse of 'the area of a cyclic quadrilateral' is '√((s−a)(s−b)(s−c)(s−d)) from the sides alone', which is also the greatest area those four sides can enclose. Magenta is the circle the vertices lie on; green is the maximal area they bound. Biggest area, from the sides.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE INVENTORY · David Lee Wise (ROOT0), with AVAN