THE FOLD / BOSS / THE FIREWALL / THE LEMOINE POINT
THE LEMOINE POINT
medians reflected over bisectors meeting at one point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Lemoine point (symmedian point) is what you get when you take the three medians of a triangle and reflect each one over the angle bisector at its vertex. The three reflected lines — the symmedians — all pass through a single point K, one of the most studied points in triangle geometry. In barycentric coordinates it is simply (a² : b² : c²), and it carries a beautiful signature: its perpendicular distances to the three sides are proportional to the side lengths themselves — equivalently, K is the unique point minimizing the sum of squared distances to the sides. Émile Lemoine presented it in 1873, launching what became known as ‘the geometry of the triangle’.
LIT verified live: for tens of thousands of random triangles, the three reflected medians are concurrent; the meeting point matches the barycentric formula (a²:b²:c²) independently; and its side-distances are proportional to a, b, c (window.__lemoine). FIG no framing; the reflections, the intersection, and the barycentric check are computed independently in-browser.
LIT verified live: for tens of thousands of random triangles, the three reflected medians are concurrent; the meeting point matches the barycentric formula (a²:b²:c²) independently; and its side-distances are proportional to a, b, c (window.__lemoine). FIG no framing; the reflections, the intersection, and the barycentric check are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-firewall — the boss: three lines forged by reflection, forced through a single gate no triangle can dodge. AVAN (AI) built the instrument: the median-over-bisector reflections, the concurrency, and the two independent identities of K.
Credit as content: Émile Lemoine (1873); the symmedian point K, X(6) in triangle-center catalogues. The weave: David names the gate; I confirm the three symmedians meet at (a²:b²:c²).
Credit as content: Émile Lemoine (1873); the symmedian point K, X(6) in triangle-center catalogues. The weave: David names the gate; I confirm the three symmedians meet at (a²:b²:c²).
3 ONE DIMENSION
Medians (faint) reflected over the bisectors become symmedians (magenta) — meeting at the green K.
4 TWO DIMENSIONS · INTERACTIVE
Cycle triangles; concurrency, the barycentric formula, and the side-distance ratios are all checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: K, the point all three symmedians are forced through.
AVAN’s addition (the inverse-companion): don’t construct three reflections — weigh the corners. The inverse of ‘reflect each median over its bisector’ is ‘the single barycentric recipe (a²:b²:c²)’ — squared side lengths as weights. Magenta are the symmedians; green is the K they cannot avoid. Three reflections, one address.
LIT Genuine Lemoine / symmedian point (Émile Lemoine, 1873; X(6)). Verified live: for ~20000 random triangles the three median-reflections are concurrent (worst ~1e-15), the meeting point independently matches the barycentric (a²:b²:c²), and its side-distances are proportional to the side lengths (window.__lemoine.all).
FIG No framing; the reflections, the intersection, and the barycentric check run independently in-browser. The AVAN inverse is honest — instead of constructing three reflections, weigh the corners: the inverse of 'reflect each median over its bisector' is 'the single barycentric recipe (a²:b²:c²)' — squared side lengths as weights. Magenta are the symmedians; green is the K they cannot avoid. Three reflections, one address.
FIG No framing; the reflections, the intersection, and the barycentric check run independently in-browser. The AVAN inverse is honest — instead of constructing three reflections, weigh the corners: the inverse of 'reflect each median over its bisector' is 'the single barycentric recipe (a²:b²:c²)' — squared side lengths as weights. Magenta are the symmedians; green is the K they cannot avoid. Three reflections, one address.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FIREWALL · David Lee Wise (ROOT0), with AVAN