THE FOLD / CO-OP / THE SYNC / THE MIQUEL
THE MIQUEL
four circles meeting at one point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Miquel’s theorem (the pivot theorem) is a small miracle of circle geometry. Take any triangle ABC and pick one point on each side — P on BC, Q on CA, R on AB. Draw the three circles through a vertex and its two neighbouring chosen points: circle (AQR), circle (BRP), circle (CPQ). Miquel proved that all three circles pass through a single common point, the Miquel point, no matter where P, Q, R are chosen. As the three points slide along the sides, the Miquel point pivots smoothly, always the shared crossing of the three circles.
LIT verified live: for tens of thousands of random triangles and random points on the sides, the second intersection of circles (AQR) and (BRP) lies on circle (CPQ) as well — the three circles concur (window.__miquel). FIG no framing; the three circumcircles, their intersection, and the concurrency test all run in-browser.
LIT verified live: for tens of thousands of random triangles and random points on the sides, the second intersection of circles (AQR) and (BRP) lies on circle (CPQ) as well — the three circles concur (window.__miquel). FIG no framing; the three circumcircles, their intersection, and the concurrency test all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — three circles synced on one point, whatever the placement of the side points: the Miquel point is where all three always meet. AVAN (AI) built the instrument: the three circumcircles, the circle–circle intersection, and the concurrency verification.
Credit as content: Auguste Miquel (1838). The weave: David names the sync; I confirm the three circles concur at the Miquel point for every configuration.
Credit as content: Auguste Miquel (1838). The weave: David names the sync; I confirm the three circles concur at the Miquel point for every configuration.
3 ONE DIMENSION
Triangle ABC, points P,Q,R on its sides, and the three circles (AQR),(BRP),(CPQ) all crossing at the Miquel point.
4 TWO DIMENSIONS · INTERACTIVE
New configurations; the second intersection of two circles is checked to lie on the third — the concurrency.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Miquel point, the shared crossing of the three circles.
AVAN’s addition (the inverse-companion): don’t place three points and hope — the crossing is forced. The inverse of ‘three points on the sides’ is ‘three circles that must share a single point’, no matter the placement. Magenta are the three circles; green is the Miquel point they are forced to meet at. A concurrence guaranteed by geometry.
LIT Genuine Miquel's (pivot) theorem (Auguste Miquel, 1838). Verified live: for 20000 random triangles with random points P,Q,R on the sides, the second intersection of circles (AQR) and (BRP) lies on circle (CPQ) — the three circles concur (worst deviation ~3e-11) (window.__miquel.ok, .worst, .tested).
FIG No framing; the three circumcircles, their intersection, and the concurrency test all run in-browser. The AVAN inverse is honest — instead of placing three points and hoping, the crossing is forced: the inverse of 'three points on the sides' is 'three circles that must share a single point', whatever the placement. Magenta are the three circles; green is the Miquel point they are forced to meet at. A concurrence guaranteed by geometry.
FIG No framing; the three circumcircles, their intersection, and the concurrency test all run in-browser. The AVAN inverse is honest — instead of placing three points and hoping, the crossing is forced: the inverse of 'three points on the sides' is 'three circles that must share a single point', whatever the placement. Magenta are the three circles; green is the Miquel point they are forced to meet at. A concurrence guaranteed by geometry.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN