F.6 Centroid

Concurrency of the Medians, Vector Average, and the Medial Triangle

The three medians AM_A, BM_B, CM_C joining the vertices of triangle ABC to the midpoints of the opposite sides concur at the centroid G = X(2), which divides each median in a 2 : 1 ratio and has homogeneous barycentric coordinates 1 : 1 : 1.

Interactive GEOMETOR construction for F.6 (Centroid). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== M_A = midpoint(B, C) M_B = midpoint(C, A) M_C = midpoint(A, B) m_a = segment(A, M_A) m_b = segment(B, M_B) m_c = segment(C, M_C) G = intersect(m_a, m_b)

Mathematical Synthesis

The centroid G \(= X(2)\) is the center of mass of \(\triangle ABC\) and the concurrency point of the three median cevians ma, mb, mc meeting the opposite side midpoints MA, MB, MC (SeeligNotebooks).

Connecting the midpoints forms the medial triangle \(\triangle M_A M_B M_C\), which is homothetic to \(\triangle ABC\) with center G and scale factor \(-1/2\), partitioning \(\triangle ABC\) into four congruent subtriangles of area \(K/4\). On the canonical \((5, 6, 7)\) triangle, G has Cartesian coordinates \((13/3, 4\sqrt{6}/7) \approx (4.3333, 1.3997)\).

Key Algebraic Identities

\[\vec{G} = \frac{\vec{A} + \vec{B} + \vec{C}}{3}, \qquad G = (1 : 1 : 1) \text{ in homogeneous barycentrics}\]

\[m_a^2 = \frac{2 b^2 + 2 c^2 - a^2}{4}, \qquad m_b^2 = \frac{2 c^2 + 2 a^2 - b^2}{4}, \qquad m_c^2 = \frac{2 a^2 + 2 b^2 - c^2}{4}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.6 Centroid — Folios (Folios 0022–0023, 2 scanned manuscript pages in resources/seelig/06_centroid)
Euclid’s Elements	I.10, VI.2 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	X(2) — Centroid (Kimberling1998)
Seelig Spreadsheets	General Reference, Centroid, 5-6-7 A,B,C (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.3 Biquadratic Forms in Heron’s Formula

Required for: F.7 Ceva and Menelaus, F.34 Nine-Point Circle, F.35 Obtuse and Acute Triangles, F.46 Stewart’s Theorem, F.52 Homogeneous Barycentric Coordinates