THE FOLD / LOOT / THE MINT / THE CONWAY CIRCLE
THE CONWAY CIRCLE
six side-extension points on one circle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Conway’s circle theorem (John Horton Conway) is a six-point surprise. Take any triangle and, at each vertex, extend the two sides beyond that vertex by the length of the side opposite it: beyond B extend both adjoining side-lines by b (= CA), beyond C by c (= AB), beyond A by a (= BC). This produces six new endpoints. The theorem: all six lie on a single circle — the Conway circle — centred at the triangle’s incentre I, with radius exactly √(r² + s²), where r is the inradius and s the semiperimeter. The proof is a one-line consequence: each extension lands a distance s from the point where the incircle touches that side.
LIT verified live: for tens of thousands of random triangles, all six extension points are equidistant from the incentre, at distance √(r²+s²), to ~1e-13 (window.__conwaycircle). FIG no framing; the incentre, r, s, and the six points are all computed independently in-browser.
LIT verified live: for tens of thousands of random triangles, all six extension points are equidistant from the incentre, at distance √(r²+s²), to ~1e-13 (window.__conwaycircle). FIG no framing; the incentre, r, s, and the six points are all computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — the loot: extend the sides by the opposite lengths and six points are minted, all on one perfect circle. AVAN (AI) built the instrument: the six side-extension points, the incentre, and the radius √(r²+s²).
Credit as content: John Horton Conway. The weave: David names the mint; I confirm the six points share the circle of radius √(r²+s²) about the incentre.
Credit as content: John Horton Conway. The weave: David names the mint; I confirm the six points share the circle of radius √(r²+s²) about the incentre.
3 ONE DIMENSION
A triangle with its sides extended by the opposite lengths — the six endpoints all land on the Conway circle.
4 TWO DIMENSIONS · INTERACTIVE
Cycle triangles; all six distances to the incentre are checked equal to √(r²+s²).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Conway circle, radius √(r²+s²) about the incentre.
AVAN’s addition (the inverse-companion): don’t track six loose endpoints — read the one circle they share. The inverse of ‘six side-extension points’ is ‘one circle of radius √(r²+s²) about the incentre’. Magenta are the six extension points; green is the circle they all sit on. Six points, one minted circle.
LIT Genuine Conway circle theorem (John Horton Conway). Verified live: for ~40000 random triangles all six side-extension points are equidistant from the incentre at radius √(r²+s²) to ~1e-13 (window.__conwaycircle.ok).
FIG No framing; the incentre, r, s, and the six points are computed independently in-browser. The AVAN inverse is honest — instead of tracking six loose endpoints, read the one circle they share: the inverse of 'six side-extension points' is 'one circle of radius √(r²+s²) about the incentre'. Magenta are the six extension points; green is the circle they all sit on. Six points, one minted circle.
FIG No framing; the incentre, r, s, and the six points are computed independently in-browser. The AVAN inverse is honest — instead of tracking six loose endpoints, read the one circle they share: the inverse of 'six side-extension points' is 'one circle of radius √(r²+s²) about the incentre'. Magenta are the six extension points; green is the circle they all sit on. Six points, one minted circle.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN