F.21 Fermat Point

Torricelli’s Equilateral Construction for X(13), 120-Degree Equiangularity, and the Obtuse Threshold

For any triangle ABC whose largest interior angle is less than 120 degrees, erecting external equilateral triangles BCX_A, CAY_B, ABZ_C on its sides yields concurrent segments AX_A, BY_B, CZ_C that intersect at the Fermat-Torricelli point F = X(13), where <AFB = <BFC = <CFA = 120 degrees and FA + FB + FC is minimized.

Interactive GEOMETOR construction for F.21 (Fermat Point). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== C’ = outer_equilateral_apex(A, B) ABC’ = polygon(A, B, C’) CC’ = segment(C, C’) F = fermat_point(A, B, C)

Mathematical Synthesis

Dated July 6, 2011 and citing Evangelista Torricelli and David Wells (Wells1991, p. 75; SeeligNotebooks), Seelig constructs the Fermat point F \(= X(13)\) by erecting external equilateral triangles (such as ABC with apex C) on the sides of \(\triangle ABC\) and intersecting their circumcircles with the Simpson/Torricelli lines CC.

In Folio 2, Seelig carefully distinguishes between the isogonic Jakob Steiner / Torricelli point (where sides subtend \(120^\circ\)) and the true distance-minimizing Fermat point when an interior angle of \(\triangle ABC\) is \(\ge 120^\circ\): in that obtuse regime, the \(120^\circ\) point moves outside \(\triangle ABC\) while the minimum of \(FA + FB + FC\) collapses onto the obtuse vertex itself.

Key Algebraic Identities

\[\angle AFB = \angle BFC = \angle CFA = 120^\circ, \qquad \min_{P} (|PA| + |PB| + |PC|) = |CC'| = \sqrt{\frac{a^2 + b^2 + c^2}{2} + 2\sqrt{3} K}\]

\[F = X(13) = \left(\csc\left(\alpha + \frac{\pi}{3}\right) : \csc\left(\beta + \frac{\pi}{3}\right) : \csc\left(\gamma + \frac{\pi}{3}\right)\right)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.21 Fermat Point — Folios (Folios 0128–0129, 2 scanned manuscript pages in resources/seelig/21_fermat-point)
Euclid’s Elements	I.1, III.21, III.22 (EuclidElements)
GEOMETOR Elements (G-Index)	G.1 point, G.3 extremities of a segment
Encyclopedia of Triangle Centers	X(13) — First Fermat (Torricelli) Point, X(14) — Second Fermat Point, X(15) — First Isodynamic Point, X(16) — Second Isodynamic Point (Kimberling1998)
Seelig Spreadsheets	General Reference, Barycentric (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.14 Conway Notation, F.18 Equilateral Triangle