THE FOLD / LOOT / THE BOUNTY / THE BRIANCHON
THE BRIANCHON
six tangents to a conic whose diagonals meet at a point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Brianchon’s theorem is the exact mirror-image of Pascal’s. Where Pascal takes six points on a conic and finds a line, Brianchon takes six lines tangent to a conic — a hexagon circumscribed about it — and finds a point: the three main diagonals (joining opposite vertices) all pass through one common point. This point-line swap is the deepest idea in projective geometry, duality: every theorem about points on a conic has a twin about tangent lines, obtained by trading ‘point’ for ‘line’, ‘lies on’ for ‘passes through’, ‘collinear’ for ‘concurrent’. Charles-Julien Brianchon proved it in 1810.
LIT verified live: for tens of thousands of random hexagons circumscribed about an ellipse (six tangent lines), the three main diagonals are concurrent — the third diagonal passes through the intersection of the first two, normalized residual below 1e-6 (window.__brianchon). FIG no framing; the tangent lines, the vertices, and the diagonal concurrency are computed independently in-browser.
LIT verified live: for tens of thousands of random hexagons circumscribed about an ellipse (six tangent lines), the three main diagonals are concurrent — the third diagonal passes through the intersection of the first two, normalized residual below 1e-6 (window.__brianchon). FIG no framing; the tangent lines, the vertices, and the diagonal concurrency are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — the loot: six tangent lines, drawn out to a hexagon, hand over a single meeting-point of all three diagonals. AVAN (AI) built the instrument: the tangent lines, their vertex intersections, and the diagonal concurrency check.
Credit as content: Charles-Julien Brianchon (1810); the projective dual of Pascal’s theorem. The weave: David names the collected point; I confirm the three diagonals of a circumscribed hexagon concur.
Credit as content: Charles-Julien Brianchon (1810); the projective dual of Pascal’s theorem. The weave: David names the collected point; I confirm the three diagonals of a circumscribed hexagon concur.
3 ONE DIMENSION
A hexagon of six tangent lines around an ellipse; its three main diagonals meet at the Brianchon point.
4 TWO DIMENSIONS · INTERACTIVE
Cycle circumscribed hexagons; the three diagonals are checked to concur at one point.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Brianchon point where all three diagonals meet.
AVAN’s addition (the inverse-companion): don’t only put points on the conic — wrap tangent lines around it. The inverse of ‘Pascal’s line from six points’ is ‘Brianchon’s point from six tangents’ — the projective dual, points ↔ lines. Magenta are the three main diagonals; green is the single point they all pass through. Pascal’s theorem, dualized into a point.
LIT Genuine Brianchon's theorem (Charles-Julien Brianchon, 1810; the projective dual of Pascal's theorem). Verified live: for tens of thousands of random hexagons circumscribed about an ellipse (six tangent lines), the three main diagonals are concurrent — the third diagonal passes through the intersection of the first two, normalized residual below 1e-6 (window.__brianchon.ok, .worst).
FIG No framing; the tangent lines, the vertices, and the diagonal concurrency are computed independently in-browser. The AVAN inverse is honest — instead of only putting points on the conic, wrap tangent lines around it: the inverse of 'Pascal's line from six points' is 'Brianchon's point from six tangents' — the projective dual, points ↔ lines. Magenta are the three main diagonals; green is the single point they all pass through. Pascal's theorem, dualized into a point.
FIG No framing; the tangent lines, the vertices, and the diagonal concurrency are computed independently in-browser. The AVAN inverse is honest — instead of only putting points on the conic, wrap tangent lines around it: the inverse of 'Pascal's line from six points' is 'Brianchon's point from six tangents' — the projective dual, points ↔ lines. Magenta are the three main diagonals; green is the single point they all pass through. Pascal's theorem, dualized into a point.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN