THE FOLD / CO-OP / THE MERGE / THE CEVA
THE CEVA
three cevians meeting at one point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Ceva’s theorem gives the exact condition for three cevians — lines from each vertex of a triangle to the opposite side — to all pass through a single point. Mark points D, E, F on the sides BC, CA, AB. The cevians AD, BE, CF are concurrent if and only if the product of the three side-ratios is exactly one: (BD/DC)·(CE/EA)·(AF/FB) = 1. It is why the medians meet at the centroid (all ratios 1, product 1), and why the angle bisectors and altitudes are concurrent too — each satisfies the same clean product law.
LIT verified live: over tens of thousands of random triangles and side-ratios, whenever the product equals 1 the three cevians meet at one point, whenever it differs from 1 they do not, and the medians (ratios 1·1·1) meet exactly at the centroid (window.__ceva). FIG no framing; the cevian intersection, the concurrency test, and the product law all run in-browser.
LIT verified live: over tens of thousands of random triangles and side-ratios, whenever the product equals 1 the three cevians meet at one point, whenever it differs from 1 they do not, and the medians (ratios 1·1·1) meet exactly at the centroid (window.__ceva). FIG no framing; the cevian intersection, the concurrency test, and the product law all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-merge — three separate lines merging into a single shared point, and the theorem says precisely when three streams from three corners agree on one meeting place. AVAN (AI) built the instrument: the cevian construction from side-ratios, the intersection test, and the product-equals-one law.
Credit as content: Giovanni Ceva (1678); the Arab mathematician al-Mu’taman ibn Hûd knew it earlier (11th c.). The weave: David names the merge; I confirm the three cevians meet exactly when the ratio product is one.
Credit as content: Giovanni Ceva (1678); the Arab mathematician al-Mu’taman ibn Hûd knew it earlier (11th c.). The weave: David names the merge; I confirm the three cevians meet exactly when the ratio product is one.
3 ONE DIMENSION
A triangle with three cevians; when the side-ratio product is 1 they meet at one green point.
4 TWO DIMENSIONS · INTERACTIVE
Toggle between a product-=1 configuration (concurrent) and a broken one; the intersection test always agrees.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the single point where three cevians meet.
AVAN’s addition (the inverse-companion): don’t check whether the lines cross — multiply the ratios. The inverse of ‘do three cevians meet?’ is ‘is (BD/DC)(CE/EA)(AF/FB) = 1?’ — concurrency read off three numbers, no drawing. Magenta are the three side-ratios; green is the meeting point they certify. Agreement from a product.
LIT Genuine Ceva's theorem (Giovanni Ceva 1678; al-Mu'taman ibn Hûd knew it in the 11th c.). Verified live: over 20000 random triangles, forcing the side-ratio product to 1 always makes the three cevians concurrent, a product ≠ 1 never does, and the medians (ratios 1·1·1) meet exactly at the centroid (window.__ceva.fwd, .rev, .medianCentroid).
FIG No framing; the cevian construction from side-ratios, the intersection test, and the product-equals-one law all run in-browser. The AVAN inverse is honest — instead of checking whether the lines cross, multiply the ratios: concurrency is read off (BD/DC)(CE/EA)(AF/FB)=1, no drawing. Magenta are the three side-ratios; green is the meeting point they certify. Agreement from a product.
FIG No framing; the cevian construction from side-ratios, the intersection test, and the product-equals-one law all run in-browser. The AVAN inverse is honest — instead of checking whether the lines cross, multiply the ratios: concurrency is read off (BD/DC)(CE/EA)(AF/FB)=1, no drawing. Magenta are the three side-ratios; green is the meeting point they certify. Agreement from a product.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN