THE FOLD / SPAWN / THE TOOLCHAIN / THE NINE-POINT CIRCLE
THE NINE-POINT CIRCLE
nine special triangle points on one circle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The nine-point circle is one of the most elegant facts about a triangle: nine special points all lie on a single circle. They are the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments from each vertex to the orthocentre. No matter how the triangle is shaped, these nine points are perfectly concyclic. The circle’s centre N is the midpoint between the circumcentre O and the orthocentre H (so N sits on the Euler line), and its radius is exactly half the circumradius, R/2. It touches the incircle and the three excircles (Feuerbach’s theorem).
LIT verified live: for tens of thousands of random triangles, all nine points are equidistant from N = midpoint(O, H), at distance exactly R/2, to ~1e-14 (window.__ninepoint). FIG no framing; the nine points, the centre, and the radius are computed independently in-browser.
LIT verified live: for tens of thousands of random triangles, all nine points are equidistant from N = midpoint(O, H), at distance exactly R/2, to ~1e-14 (window.__ninepoint). FIG no framing; the nine points, the centre, and the radius are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the spawn: nine points from three different constructions compile, every time, onto one circle of radius R/2. AVAN (AI) built the instrument: the side-midpoints, altitude-feet, Euler-point midpoints, and the shared circle.
Credit as content: Poncelet and Brianchon (the nine-point circle); Karl Feuerbach (the tangency). The weave: David names the compile; I confirm all nine points sit on the circle of radius R/2 about the midpoint of O and H.
Credit as content: Poncelet and Brianchon (the nine-point circle); Karl Feuerbach (the tangency). The weave: David names the compile; I confirm all nine points sit on the circle of radius R/2 about the midpoint of O and H.
3 ONE DIMENSION
A triangle with its nine points — side midpoints, altitude feet, Euler-point midpoints — all on one circle.
4 TWO DIMENSIONS · INTERACTIVE
Cycle triangles; all nine distances to the centre N are checked equal to R/2.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the nine-point circle, radius R/2 about the midpoint of O and H.
AVAN’s addition (the inverse-companion): don’t track nine points from three constructions — read the one circle they share. The inverse of ‘nine special points’ is ‘a single circle of radius R/2 centred midway between circumcentre and orthocentre’. Magenta are the nine points; green is the circle carrying them all. Three constructions, one circle.
LIT Genuine nine-point circle (Poncelet and Brianchon; Feuerbach for the tangency). Verified live: for ~40000 random triangles, all nine points (three side-midpoints, three altitude-feet, three Euler-point midpoints) are equidistant from N = midpoint(O, H) at distance exactly R/2, to ~1e-14 (window.__ninepoint.ok, .worst).
FIG No framing; the nine points, the centre, and the radius are computed independently in-browser. The AVAN inverse is honest — instead of tracking nine points from three constructions, read the one circle they share: the inverse of 'nine special points' is 'a single circle of radius R/2 centred midway between circumcentre and orthocentre'. Magenta/gold/cyan are the nine points; green is the circle carrying them all. Three constructions, one circle.
FIG No framing; the nine points, the centre, and the radius are computed independently in-browser. The AVAN inverse is honest — instead of tracking nine points from three constructions, read the one circle they share: the inverse of 'nine special points' is 'a single circle of radius R/2 centred midway between circumcentre and orthocentre'. Magenta/gold/cyan are the nine points; green is the circle carrying them all. Three constructions, one circle.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN