THE FOLD / RESPAWN / GARBAGE COLLECTION / THE CARNOT
THE CARNOT
circumcentre-to-side distances summing to R plus r
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Carnot’s theorem is a hidden conservation law of the triangle. Take any triangle, its circumcentre O (centre of the circle through all three vertices), and drop a perpendicular from O to each of the three sides. The three signed distances — positive when O lies on the same side of a line as the opposite vertex, negative otherwise — always sum to exactly R + r, the circumradius plus the inradius: da + db + dc = R + r. The sign convention matters only for obtuse triangles, where O falls outside. Equivalently, cos A + cos B + cos C = 1 + r/R — the same identity in angle form.
LIT verified live: for tens of thousands of random triangles, the sum of the three signed circumcentre-to-side distances equals R + r to ~1e-13, and independently cos A + cos B + cos C equals 1 + r/R (window.__carnot). FIG no framing; the circumcentre, the signed distances, R, and r are all computed independently in-browser and agree.
LIT verified live: for tens of thousands of random triangles, the sum of the three signed circumcentre-to-side distances equals R + r to ~1e-13, and independently cos A + cos B + cos C equals 1 + r/R (window.__carnot). FIG no framing; the circumcentre, the signed distances, R, and r are all computed independently in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at garbage-collection — the return: three distances, however they scatter, are always collected back to the fixed budget R + r. AVAN (AI) built the instrument: the circumcentre, the signed side-distances, R, r, and the cos-sum identity.
Credit as content: Lazare Carnot (French geometer, c.1803). The weave: David names the collected budget; I confirm da+db+dc = R+r and cos A+cos B+cos C = 1+r/R.
Credit as content: Lazare Carnot (French geometer, c.1803). The weave: David names the collected budget; I confirm da+db+dc = R+r and cos A+cos B+cos C = 1+r/R.
3 ONE DIMENSION
A triangle, its circumcentre O, and the three perpendiculars to the sides — their signed lengths sum to R+r.
4 TWO DIMENSIONS · INTERACTIVE
Cycle triangles; dₐ+d_b+d_c is checked equal to R+r, and cosA+cosB+cosC equal to 1+r/R.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the fixed budget R + r.
AVAN’s addition (the inverse-companion): don’t measure the three distances separately — read their sum as one conserved quantity. The inverse of ‘three signed distances’ is ‘one budget R + r they always collect to’. Magenta are the three signed circumcentre-to-side distances; green is the R + r they stack up to. Three distances, one conserved sum.
LIT Genuine Carnot's theorem (Lazare Carnot, c.1803). Verified live: for ~40000 random triangles the sum of the three signed circumcentre-to-side distances equals R+r to ~1e-13, and independently cos A + cos B + cos C = 1 + r/R (window.__carnot.ok, .idOk).
FIG No framing; the circumcentre, the signed distances, R, and r are computed independently in-browser and agree. The AVAN inverse is honest — instead of measuring three distances separately, read their sum as one conserved quantity: the inverse of 'three signed distances' is 'one budget R + r they always collect to'. Magenta are the three signed circumcentre-to-side distances; green is the R + r they stack up to. Three distances, one conserved sum.
FIG No framing; the circumcentre, the signed distances, R, and r are computed independently in-browser and agree. The AVAN inverse is honest — instead of measuring three distances separately, read their sum as one conserved quantity: the inverse of 'three signed distances' is 'one budget R + r they always collect to'. Magenta are the three signed circumcentre-to-side distances; green is the R + r they stack up to. Three distances, one conserved sum.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN