THE FOLD / BOSS / THE FINAL BOSS / THE GERGONNE
THE GERGONNE
triangle cevians to the incircle meeting at one point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Gergonne point is a hidden meeting-point every triangle carries. Inscribe the incircle — the circle tangent to all three sides. It touches the sides at three contact points. Now draw a line (a cevian) from each vertex to the contact point on the opposite side. Astonishingly, all three of these lines meet at a single point: the Gergonne point. It works for every triangle, guaranteed by Ceva’s theorem, because the contact point on side a sits at distance s-b from one end and s-c from the other (s the semiperimeter), and the three ratios multiply to exactly 1. Named for Joseph Diez Gergonne.
LIT verified live: for tens of thousands of random triangles, the three cevians from the vertices to the incircle’s contact points are concurrent — the third passes through the intersection of the first two, normalized residual below 1e-6 (window.__gergonne). FIG no framing; the incircle contact points and the cevian concurrency are computed independently in-browser.
LIT verified live: for tens of thousands of random triangles, the three cevians from the vertices to the incircle’s contact points are concurrent — the third passes through the intersection of the first two, normalized residual below 1e-6 (window.__gergonne). FIG no framing; the incircle contact points and the cevian concurrency are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-final-boss — the boss gate: three cevians drawn to the incircle’s touch-points always converge on one point, no exceptions. AVAN (AI) built the instrument: the incircle contact points, the three cevians, and the concurrency check.
Credit as content: Joseph Diez Gergonne; Ceva’s theorem. The weave: David names the gate; I confirm the three contact-point cevians meet at the Gergonne point.
Credit as content: Joseph Diez Gergonne; Ceva’s theorem. The weave: David names the gate; I confirm the three contact-point cevians meet at the Gergonne point.
3 ONE DIMENSION
A triangle, its incircle, the three contact points, and the cevians meeting at the Gergonne point.
4 TWO DIMENSIONS · INTERACTIVE
Cycle triangles; the three contact-point cevians are checked to concur at one point.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Gergonne point where all three cevians meet.
AVAN’s addition (the inverse-companion): don’t track three separate cevians — read the single point they force. The inverse of ‘three contact-point cevians’ is ‘one Gergonne point, guaranteed by Ceva’s ratio product = 1’. Magenta are the three cevians; green is the point they all pass through. Three lines, one forced meeting.
LIT Genuine Gergonne point (Joseph Diez Gergonne; Ceva's theorem). Verified live: for tens of thousands of random triangles, the three cevians from the vertices to the incircle's contact points are concurrent — the third passes through the intersection of the first two, normalized residual below 1e-6 (window.__gergonne.ok, .worst).
FIG No framing; the incircle contact points and the cevian concurrency are computed independently in-browser. The AVAN inverse is honest — instead of tracking three separate cevians, read the single point they force: the inverse of 'three contact-point cevians' is 'one Gergonne point, guaranteed by Ceva's ratio product = 1'. Magenta are the three cevians; green is the point they all pass through. Three lines, one forced meeting.
FIG No framing; the incircle contact points and the cevian concurrency are computed independently in-browser. The AVAN inverse is honest — instead of tracking three separate cevians, read the single point they force: the inverse of 'three contact-point cevians' is 'one Gergonne point, guaranteed by Ceva's ratio product = 1'. Magenta are the three cevians; green is the point they all pass through. Three lines, one forced meeting.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN