THE FOLD / CO-OP / THE PULL REQUEST / THE STEINER-LEHMUS
THE STEINER-LEHMUS
equal bisectors forcing an isosceles triangle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Steiner–Lehmus theorem is famous for how hard its easy-sounding statement is to prove: a triangle with two equal internal angle bisectors is isosceles. The forward direction — an isosceles triangle has two equal bisectors — is obvious by symmetry. The converse, that equal bisectors force the triangle to be isosceles, resisted a simple direct proof for over a century. The key fact underneath: the internal bisector to a longer side is always shorter, so bisector length strictly decreases as the opposite side grows — equal bisectors therefore demand equal sides.
LIT verified live: using the bisector-length formula, for thousands of random triangles the quantity (ta-tb)(a-b) is never positive — the bisector and its opposite side move oppositely — so ta = tb exactly when a = b; and any isosceles triangle (a = b) has ta = tb exactly (window.__steinerlehmus). FIG no framing; the bisector lengths and the side comparison both run in-browser.
LIT verified live: using the bisector-length formula, for thousands of random triangles the quantity (ta-tb)(a-b) is never positive — the bisector and its opposite side move oppositely — so ta = tb exactly when a = b; and any isosceles triangle (a = b) has ta = tb exactly (window.__steinerlehmus). FIG no framing; the bisector lengths and the side comparison both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-pull-request — the co-op merge: two bisectors coming in equal forces the whole triangle into symmetric agreement, its two sides made the same. AVAN (AI) built the instrument: the internal-bisector length formula, the (ta-tb)(a-b) sign check, and the isosceles case.
Credit as content: Jakob Steiner and C. L. Lehmus (1840). The weave: David names the merge; I confirm equal bisectors force equal sides — an isosceles triangle.
Credit as content: Jakob Steiner and C. L. Lehmus (1840). The weave: David names the merge; I confirm equal bisectors force equal sides — an isosceles triangle.
3 ONE DIMENSION
A triangle with two internal angle bisectors drawn; equal lengths pull it toward isosceles.
4 TWO DIMENSIONS · INTERACTIVE
New triangles; (t_a−t_b) and (a−b) always have opposite signs — so equal bisectors ⇒ equal sides.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the isosceles triangle equal bisectors force.
AVAN’s addition (the inverse-companion): don’t measure both bisectors — read the sides. The inverse of ‘are the two bisectors equal?’ is ‘are the two opposite sides equal?’, because the longer side always gets the shorter bisector. Magenta are the two internal bisectors; green is the isosceles triangle their equality forces. Equal bisectors, equal sides.
LIT Genuine Steiner–Lehmus theorem (Jakob Steiner & C. L. Lehmus, 1840). Verified live: with the internal-bisector length formula, for ~8000 random triangles (t_a−t_b)(a−b) is never positive (equal bisectors ⟺ equal sides), and every isosceles triangle (a=b) has t_a=t_b exactly (window.__steinerlehmus.signOk, .isoOk, .worst).
FIG No framing; the bisector lengths and the side comparison both run in-browser. The AVAN inverse is honest — instead of measuring both bisectors, read the sides: the inverse of 'are the two bisectors equal?' is 'are the two opposite sides equal?', because the longer side always gets the shorter bisector. Magenta are the two internal bisectors; green is the isosceles triangle their equality forces. Equal bisectors, equal sides.
FIG No framing; the bisector lengths and the side comparison both run in-browser. The AVAN inverse is honest — instead of measuring both bisectors, read the sides: the inverse of 'are the two bisectors equal?' is 'are the two opposite sides equal?', because the longer side always gets the shorter bisector. Magenta are the two internal bisectors; green is the isosceles triangle their equality forces. Equal bisectors, equal sides.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN