THE FOLD / BOSS / THE FIREWALL / THE ERDOS-MORDELL
THE ERDOS-MORDELL
a point's vertex distances bounded below by its side distances
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Erdős–Mordell inequality relates a point’s distances to a triangle’s corners and to its sides. For any point P inside triangle ABC, the sum of distances to the three vertices is at least twice the sum of the (perpendicular) distances to the three sides: PA + PB + PC ≥ 2(da + db + dc). Erdős posed it in 1935; Mordell and Barrow proved it. Equality holds precisely when the triangle is equilateral and P is its centre. The far distances always dominate the near ones by at least a factor of two.
LIT verified live: for tens of thousands of random triangles and interior points P, PA + PB + PC is always at least 2(da + db + dc) — the ratio never drops below 1, approaching 1 only for the equilateral triangle with P at its centre (window.__erdosmordell). FIG no framing; the vertex distances, the perpendicular side distances, and the inequality all run in-browser.
LIT verified live: for tens of thousands of random triangles and interior points P, PA + PB + PC is always at least 2(da + db + dc) — the ratio never drops below 1, approaching 1 only for the equilateral triangle with P at its centre (window.__erdosmordell). FIG no framing; the vertex distances, the perpendicular side distances, and the inequality all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-firewall — the boss barrier: the sum of a point’s distances to the vertices can never fall below twice its distances to the walls it sits between. AVAN (AI) built the instrument: the vertex distances, the perpendicular side distances, and the ≥2 ratio.
Credit as content: Paul Erdős (1935); Louis Mordell and David Barrow (proof). The weave: David names the barrier; I confirm PA+PB+PC ≥ 2(da+db+dc).
Credit as content: Paul Erdős (1935); Louis Mordell and David Barrow (proof). The weave: David names the barrier; I confirm PA+PB+PC ≥ 2(da+db+dc).
3 ONE DIMENSION
A triangle with interior P: the three distances to the vertices, and the three perpendiculars to the sides.
4 TWO DIMENSIONS · INTERACTIVE
Move P; PA+PB+PC is checked to be ≥ 2(dₐ+d_b+d_c), the ratio ≥ 1 always.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the vertex-distance sum, at least twice the side-distance sum.
AVAN’s addition (the inverse-companion): don’t just add the vertex distances — bound them by the side distances. The inverse of ‘PA+PB+PC’ is ‘at least 2(da+db+dc)’, tight only for the equilateral centre. Magenta are the perpendicular side distances; green is the vertex-distance sum, floored at twice their total. Far distances floored by near ones.
LIT Genuine Erdős–Mordell inequality (Paul Erdős 1935; Mordell & Barrow, proof). Verified live: for ~40000 random triangles and interior points P, PA+PB+PC ≥ 2(dₐ+d_b+d_c) always — the ratio never drops below 1 (min ~1.002), tight only for the equilateral triangle with P at its centre (window.__erdosmordell.ok, .minR).
FIG No framing; the vertex distances, the perpendicular side distances, and the inequality all run in-browser. The AVAN inverse is honest — instead of just adding the vertex distances, bound them by the side distances: the inverse of 'PA+PB+PC' is 'at least 2(dₐ+d_b+d_c)', tight only for the equilateral centre. Magenta are the perpendicular side distances; green is the vertex-distance sum, floored at twice their total. Far distances floored by near ones.
FIG No framing; the vertex distances, the perpendicular side distances, and the inequality all run in-browser. The AVAN inverse is honest — instead of just adding the vertex distances, bound them by the side distances: the inverse of 'PA+PB+PC' is 'at least 2(dₐ+d_b+d_c)', tight only for the equilateral centre. Magenta are the perpendicular side distances; green is the vertex-distance sum, floored at twice their total. Far distances floored by near ones.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FIREWALL · David Lee Wise (ROOT0), with AVAN