F.18 Equilateral Triangle

Coincidence of Centers, h_A = 3r, and Complex Cube-Root Proof of an Incenter-Bisector Equilateral Triangle

In an equilateral triangle of side a, the incenter, centroid, circumcenter, and orthocenter coincide, with altitude h_A = (sqrt(3)/2)a = 3r and R = 2r; moreover, points A’, B’, C’ placed along the angle bisectors AI, BI, CI of any triangle ABC at distances A’I = (2r/sqrt(3)) sin(alpha/2 + pi/6) form an equilateral triangle.

Interactive GEOMETOR construction for F.18 (Equilateral Triangle). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point ===== c_a = circle(A, B) c_b = circle(B, A) C, D = intersect(c_a, c_b) ABC = polygon(A, B, C)

Mathematical Synthesis

The equilateral triangle constructed in EuclidElements Proposition I.1 (G.1) is the seed of constructive geometry: intersecting congruent circles AB and BA produces apex C with altitude \(h_A = \frac{\sqrt{3}}{2}a = 3r\) and \(R = 2r\), where G, \(I\), \(O\), and \(H\) all coincide (SeeligNotebooks).

In Folios 2–3 (equilateral-2-001–002), Seelig gives an algebraic and complex-number proof that points \(A', B', C'\) chosen on the interior angle bisectors \(AI, BI, CI\) of any triangle \(\triangle ABC\) with \(A'I = \frac{2r}{\sqrt{3}}\sin(\frac{\alpha}{2} + \frac{\pi}{6})\) form an equilateral triangle \(\triangle A'B'C'\) by verifying \(m + n\epsilon + p\epsilon^2 = 0\) for \(\epsilon = e^{i 2\pi/3}\).

Key Algebraic Identities

\[h_A = \frac{\sqrt{3}}{2} a = 3 r, \qquad R = \frac{a}{\sqrt{3}} = 2 r, \qquad K = \frac{\sqrt{3}}{4} a^2\]

\[A'I = \frac{2 r}{\sqrt{3}}\sin\left(\frac{\alpha}{2} + \frac{\pi}{6}\right), \qquad P(\epsilon) = m + n\epsilon + p\epsilon^2 = 0 \quad (\epsilon = e^{i 2\pi/3})\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.18 Equilateral Triangle — Folios (Folios 0122–0124, 3 scanned manuscript pages in resources/seelig/18_equilateral-triangle)
Euclid’s Elements	I.1, IV.2, IV.3 (EuclidElements)
GEOMETOR Elements (G-Index)	G.1 point, G.2 line
Encyclopedia of Triangle Centers	X(1) — Incenter, X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter (Kimberling1998)
Seelig Spreadsheets	General Reference, Polygon Table (Spreadsheets)

Constructive Dependency Graph

Required for: F.21 Fermat Point, F.23 Golden Triangles, F.24 Gothic Arch, F.45 Squares, Subdivided Lattices, and the Japanese Theorem, F.49 Vesica Piscis and Acute-Obtuse Probability