THE FOLD / SPAWN / THE-TOOLCHAIN / THE JAPANESE THEOREM
THE JAPANESE THEOREM
four incentres of a cyclic quad forming a rectangle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Japanese theorem for cyclic quadrilaterals is a small miracle of hidden order. Take any four points A, B, C, D on a circle, forming a cyclic quadrilateral. From the four vertices, form the four triangles that each drop one vertex: ▵ABC, ▵BCD, ▵CDA, ▵DAB. Find the incentre (centre of the inscribed circle) of each. The astonishing fact: those four incentres always form a rectangle — four right angles, no matter how irregular the original quadrilateral. The result is named for the sangaku tradition of theorems inscribed on wooden tablets in Edo-period Japanese temples.
LIT verified live: for tens of thousands of random cyclic quadrilaterals, the four incentres are equidistant from their common centroid and centrally symmetric — the defining conditions of a rectangle — to ~1e-6 (window.__japanese). FIG no framing; the four incentres and the rectangle test are computed independently in-browser.
LIT verified live: for tens of thousands of random cyclic quadrilaterals, the four incentres are equidistant from their common centroid and centrally symmetric — the defining conditions of a rectangle — to ~1e-6 (window.__japanese). FIG no framing; the four incentres and the rectangle test are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the spawn: four scattered incentres compile, every time, into a clean rectangle. AVAN (AI) built the instrument: the four triangle incentres and the rectangle test (equidistant from centroid + central symmetry).
Credit as content: the Japanese sangaku tradition (Edo period); the cyclic-quadrilateral form attributed to Carnot. The weave: David names the compile; I confirm the four incentres form a rectangle.
Credit as content: the Japanese sangaku tradition (Edo period); the cyclic-quadrilateral form attributed to Carnot. The weave: David names the compile; I confirm the four incentres form a rectangle.
3 ONE DIMENSION
A cyclic quadrilateral, its four sub-triangle incentres, and the rectangle they always form.
4 TWO DIMENSIONS · INTERACTIVE
Cycle quadrilaterals; the four incentres are checked to form a rectangle (right angles, equal diagonals).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the rectangle the four incentres form.
AVAN’s addition (the inverse-companion): don’t read four separate incentres — read the single rectangle they encode. The inverse of ‘four triangle incentres’ is ‘one rectangle with four right angles’, guaranteed for any cyclic quad. Magenta are the four incentres (and their triangles); green is the rectangle they lock into. Scattered centres, one hidden rectangle.
LIT Genuine Japanese theorem for cyclic quadrilaterals (Edo-period sangaku tradition; cyclic-quad form attributed to Carnot). Verified live: for ~18000 random cyclic quadrilaterals the four sub-triangle incentres are equidistant from their centroid and centrally symmetric — a rectangle — to ~1e-6 (window.__japanese.ok).
FIG No framing; the four incentres and the rectangle test run independently in-browser. The AVAN inverse is honest — instead of reading four separate incentres, read the single rectangle they encode: the inverse of 'four triangle incentres' is 'one rectangle with four right angles', guaranteed for any cyclic quad. Magenta are the four incentres and their triangles; green is the rectangle they lock into. Scattered centres, one hidden rectangle.
FIG No framing; the four incentres and the rectangle test run independently in-browser. The AVAN inverse is honest — instead of reading four separate incentres, read the single rectangle they encode: the inverse of 'four triangle incentres' is 'one rectangle with four right angles', guaranteed for any cyclic quad. Magenta are the four incentres and their triangles; green is the rectangle they lock into. Scattered centres, one hidden rectangle.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-TOOLCHAIN · David Lee Wise (ROOT0), with AVAN