THE FOLD / SPAWN / NULL ISLAND / THE EULER LINE
THE EULER LINE
three triangle centres that always fall on one line
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Euler line is a quiet miracle of triangle geometry: for any triangle, three of its most important centres — the centroid G (where the medians cross), the circumcenter O (centre of the circle through the vertices), and the orthocenter H (where the altitudes meet) — always lie on a single straight line. And their spacing is fixed: G sits between O and H, dividing the segment so that OG : GH = 1 : 2. Three centres, defined in utterly different ways, forever collinear in the same proportion. Euler proved it in 1765.
LIT verified live: for thousands of random triangles, O, G, H are collinear (their cross product vanishes) and H − G = 2(G − O), the 1 : 2 ratio (window.__eulerline). FIG no framing; the three centres computed and their collinearity and spacing checked.
LIT verified live: for thousands of random triangles, O, G, H are collinear (their cross product vanishes) and H − G = 2(G − O), the 1 : 2 ratio (window.__eulerline). FIG no framing; the three centres computed and their collinearity and spacing checked.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at null-island — three centres spawned from a triangle by wholly different constructions, yet born onto one line in a fixed 1 : 2 spacing. That shared origin is the boot. AVAN (AI) built the instrument: the centroid average, the circumcenter from perpendicular bisectors, the orthocenter H = A + B + C − 2O, and the collinearity and ratio checks.
Credit as content: Leonhard Euler (1765). The weave: David names null-island; I find the three centres of a triangle — averaging the vertices, solving for the equidistant point, and placing the orthocenter — and confirm they always fall on one line with G twice as close to O as to H.
Credit as content: Leonhard Euler (1765). The weave: David names null-island; I find the three centres of a triangle — averaging the vertices, solving for the equidistant point, and placing the orthocenter — and confirm they always fall on one line with G twice as close to O as to H.
3 ONE DIMENSION
Centroid G, circumcenter O, orthocenter H — always collinear, with OG : GH = 1 : 2. (In an equilateral triangle all three coincide; otherwise they string out along the Euler line.)
4 TWO DIMENSIONS · INTERACTIVE
A triangle, its three centres, and the Euler line through them; the collinearity and 1 : 2 ratio checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: three centres on one line.
AVAN’s addition (the inverse-companion): don’t treat a triangle’s centres as unrelated points — they are locked to one line in a fixed 1 : 2 ratio, so any two determine the third. The inverse of ‘construct each centre separately’ is ‘the Euler line ties them together: H = 3G − 2O.’ Magenta is the trio seen as scattered centres; green is the single Euler line binding them. Different constructions, one line.
LIT Genuine Euler line (Leonhard Euler, 1765). Verified live: for 3000 random triangles the centroid, circumcenter (from perpendicular bisectors), and orthocenter (H = A+B+C−2O) are collinear — their pairwise cross product vanishes (window.__eulerline.collinear) — and H − G = 2(G − O), the OG : GH = 1 : 2 ratio (window.__eulerline.ratio).
FIG No framing: the centroid average, the circumcenter solve, the orthocenter placement, and the collinearity and ratio checks all run in-browser with exact coordinate arithmetic. The AVAN inverse is honest — the three centres are locked to one line in a fixed 1:2 ratio (H = 3G − 2O), so any two determine the third; magenta is the trio seen as scattered, green the single Euler line binding them. Different constructions, one line.
FIG No framing: the centroid average, the circumcenter solve, the orthocenter placement, and the collinearity and ratio checks all run in-browser with exact coordinate arithmetic. The AVAN inverse is honest — the three centres are locked to one line in a fixed 1:2 ratio (H = 3G − 2O), so any two determine the third; magenta is the trio seen as scattered, green the single Euler line binding them. Different constructions, one line.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN