THE FOLD / BOSS / THE GATEKEEPER / THE SIMSON LINE
THE SIMSON LINE
feet that align only on the circle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Simson line: drop perpendiculars from a point P to the three sides of a triangle and mark the three feet. In general those feet form a little triangle — but the instant P lands on the circumcircle, the three feet fall exactly on one straight line (the Simson line of P). And it is an if-and-only-if: the feet are collinear precisely when P is on the circle. The pedal triangle’s area is proportional to |R² − OP²|, which is zero exactly on the circle.
LIT verified live: for thousands of points P on the circumcircle the three feet are collinear to machine precision, while points off the circle give a pedal triangle of clearly non-zero area (window.__simson). FIG no framing; circumcircle, feet, and their collinearity are all computed in-browser.
LIT verified live: for thousands of points P on the circumcircle the three feet are collinear to machine precision, while points off the circle give a pedal triangle of clearly non-zero area (window.__simson). FIG no framing; circumcircle, feet, and their collinearity are all computed in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-gatekeeper — the circumcircle is the gate: stand on it and the three feet snap into a line; step off and the line breaks. AVAN (AI) built the instrument: compute the circumcircle, drop the three perpendicular feet from P, and measure whether they are collinear as P moves on and off the circle.
Credit as content: the Simson–Wallace line (attributed to Robert Simson; first published by William Wallace, 1799). The weave: David names the gatekeeper; I confirm the feet are collinear if and only if P is on the circumcircle.
Credit as content: the Simson–Wallace line (attributed to Robert Simson; first published by William Wallace, 1799). The weave: David names the gatekeeper; I confirm the feet are collinear if and only if P is on the circumcircle.
3 ONE DIMENSION
A triangle, its circumcircle, a point P on the circle, the three perpendicular feet, and the single Simson line through them.
4 TWO DIMENSIONS · INTERACTIVE
Move P around the circle (collinear feet) or push it off (a real pedal triangle). The collinearity residual reads zero exactly on the circle.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Simson line, when P sits on the circle.
AVAN’s addition (the inverse-companion): don’t ask where the feet land — ask what makes them collapse. The inverse of ‘drop the feet and see’ is ‘the pedal triangle’s area is |R²−OP²|-proportional, so it vanishes exactly on the circle.’ Magenta is the pedal triangle off the circle; green is the line it collapses to on it. The circle is the zero set.
LIT Genuine Simson–Wallace line (attributed to Robert Simson; published by William Wallace, 1799): the feet of the perpendiculars from P to a triangle's sides are collinear iff P lies on the circumcircle. Verified live: on-circle feet collinear to <1e-6 relative area, off-circle relative pedal area ≥0.18 (window.__simson.{onCircleCollinear,offCircleNot}).
FIG No framing: the circumcircle, the three feet, and their collinearity are computed in-browser as P moves on and off the circle. The AVAN inverse is honest — the pedal-triangle area being |R²−OP²|-proportional (hence zero exactly on the circle) is the reason the feet collapse to a line; magenta is the pedal triangle off the circle, green the Simson line on it. The circle is the zero set.
FIG No framing: the circumcircle, the three feet, and their collinearity are computed in-browser as P moves on and off the circle. The AVAN inverse is honest — the pedal-triangle area being |R²−OP²|-proportional (hence zero exactly on the circle) is the reason the feet collapse to a line; magenta is the pedal triangle off the circle, green the Simson line on it. The circle is the zero set.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GATEKEEPER · David Lee Wise (ROOT0), with AVAN