THE FOLD / BOSS / THE RAID / THE FEUERBACH
THE FEUERBACH
a nine-point circle tangent to the incircle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Feuerbach’s theorem is one of the most beautiful coincidences in triangle geometry. Every triangle has a nine-point circle — the circle passing through nine special points (the three side midpoints, the three altitude feet, and the three midpoints from the orthocenter to the vertices), with radius exactly half the circumradius. Feuerbach proved that this nine-point circle is tangent to the incircle (and to all three excircles). The single point where it touches the incircle is the celebrated Feuerbach point. Tangency means the distance between the two circles’ centres equals the difference of their radii: |N₉ - I| = R/2 - r.
LIT verified live: for thousands of random triangles, the distance between the nine-point centre and the incentre equals R/2 - r (the nine-point radius minus the inradius) to ~1e-15 — confirming the internal tangency of the two circles (window.__feuerbach). FIG no framing; the nine-point circle, the incircle, and the tangency condition all run in-browser.
LIT verified live: for thousands of random triangles, the distance between the nine-point centre and the incentre equals R/2 - r (the nine-point radius minus the inradius) to ~1e-15 — confirming the internal tangency of the two circles (window.__feuerbach). FIG no framing; the nine-point circle, the incircle, and the tangency condition all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-raid — the boss reveal: two circles built from utterly different constructions of a triangle turn out to kiss at a single point. AVAN (AI) built the instrument: the nine-point circle (centre and R/2 radius), the incircle, and the tangency test.
Credit as content: Karl Wilhelm Feuerbach (1822). The weave: David names the reveal; I confirm the nine-point circle is tangent to the incircle at the Feuerbach point.
Credit as content: Karl Wilhelm Feuerbach (1822). The weave: David names the reveal; I confirm the nine-point circle is tangent to the incircle at the Feuerbach point.
3 ONE DIMENSION
A triangle, its nine-point circle and its incircle — tangent at the single Feuerbach point.
4 TWO DIMENSIONS · INTERACTIVE
New triangles; the distance between nine-point centre and incentre is checked to equal R/2 − r.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the tangency of the nine-point circle and the incircle.
AVAN’s addition (the inverse-companion): don’t track nine points — know the tangency. The inverse of ‘the nine-point circle’ is ‘a circle of radius R/2 that touches the incircle from outside’, their centres exactly R/2 - r apart. Magenta are the nine-point circle and incircle; green is the Feuerbach point where they touch. Two circles forced to kiss.
LIT Genuine Feuerbach's theorem (Karl Wilhelm Feuerbach, 1822). Verified live: for ~4000 random triangles, the distance between the nine-point centre and the incentre equals R/2−r (nine-point radius minus inradius) to ~1e-15, confirming the internal tangency (window.__feuerbach.ok, .worst, .n).
FIG No framing; the nine-point circle, the incircle, and the tangency condition all run in-browser. The AVAN inverse is honest — instead of tracking nine points, know the tangency: the inverse of 'the nine-point circle' is 'a circle of radius R/2 that touches the incircle', their centres exactly R/2−r apart. Magenta are the nine-point circle and incircle; green is the Feuerbach point where they touch. Two circles forced to kiss.
FIG No framing; the nine-point circle, the incircle, and the tangency condition all run in-browser. The AVAN inverse is honest — instead of tracking nine points, know the tangency: the inverse of 'the nine-point circle' is 'a circle of radius R/2 that touches the incircle', their centres exactly R/2−r apart. Magenta are the nine-point circle and incircle; green is the Feuerbach point where they touch. Two circles forced to kiss.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN