THE FOLD / CHEAT / THE SHORTCUT / THE ROUTH
THE ROUTH
a cevian triangle's area is a closed form
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Routh’s theorem gives the area of the little triangle three cevians carve out of a big one — as an exact closed form. Draw cevians from each vertex cutting the opposite side in ratios x, y, z (BD/DC = x, CE/EA = y, AF/FB = z). The three cevians bound a central triangle, and its area as a fraction of the whole is (xyz − 1)² / [(xy + x + 1)(yz + y + 1)(zx + z + 1)]. When x = y = z = 2 the fraction is exactly 1/7 — the famous one-seventh-area triangle. When xyz = 1 the cevians are concurrent (Ceva) and the triangle vanishes to a point.
LIT verified live: for thousands of random ratio triples, the closed form matches the directly-constructed central-triangle area to machine precision, and x=y=z=2 gives 1/7 (window.__routh). FIG no framing; both the geometric construction and the formula run in-browser.
LIT verified live: for thousands of random ratio triples, the closed form matches the directly-constructed central-triangle area to machine precision, and x=y=z=2 gives 1/7 (window.__routh). FIG no framing; both the geometric construction and the formula run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-shortcut — instead of intersecting three cevians and measuring the middle, take the shortcut: one formula in x, y, z. AVAN (AI) built the instrument: place a triangle, drop the three cevians, intersect them for the central triangle, measure its area by the shoelace rule, and compare to Routh’s expression.
Credit as content: Edward John Routh (1896). The weave: David names the shortcut; I construct the long way (three intersections and an area) and confirm it equals the closed form — including the surprising 1/7 at ratio 2.
Credit as content: Edward John Routh (1896). The weave: David names the shortcut; I construct the long way (three intersections and an area) and confirm it equals the closed form — including the surprising 1/7 at ratio 2.
3 ONE DIMENSION
The classic case: cevians at ratio 2 on every side cut out a central triangle of exactly one-seventh the area.
4 TWO DIMENSIONS · INTERACTIVE
Change the three ratios; the central triangle redraws, and its measured area fraction tracks Routh’s formula exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the central triangle the three cevians actually enclose.
AVAN’s addition (the inverse-companion): don’t intersect and measure — read the area straight off x, y, z. The inverse of ‘construct the triangle to find its area’ is ‘the area is a function of the three ratios alone.’ Magenta is the laborious construction; green is the one-line answer. The ratios already know the area.
LIT Genuine Routh's theorem (Edward John Routh, 1896): central cevian-triangle area / whole = (xyz−1)²/[(xy+x+1)(yz+y+1)(zx+z+1)]. Verified live: closed form matches shoelace-measured construction over 4000 random triples to <1e-7 (window.__routh.matches), and x=y=z=2 gives exactly 1/7 (window.__routh.oneSeventh).
FIG No framing: both the geometric construction (three cevian intersections, shoelace area) and the formula run in-browser. The AVAN inverse is honest — reading the area straight off the three ratios rather than constructing and measuring the triangle is exactly the theorem's shortcut; magenta is the laborious construction, green the one-line answer. The ratios already know the area.
FIG No framing: both the geometric construction (three cevian intersections, shoelace area) and the formula run in-browser. The AVAN inverse is honest — reading the area straight off the three ratios rather than constructing and measuring the triangle is exactly the theorem's shortcut; magenta is the laborious construction, green the one-line answer. The ratios already know the area.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SHORTCUT · David Lee Wise (ROOT0), with AVAN