F.46 Stewart’s Theorem

Cevian Length Identity f^2 = k a^2 + k’ b^2 - k k’ c^2 and Apollonius’s Median Formula

If a cevian of length f from C divides base c = AB into segments k’ c and k c (with k + k’ = 1), Stewart’s Theorem states f^2 = k a^2 + k’ b^2 - k k’ c^2; at the midpoint k = k’ = 1/2, this yields Apollonius’s median formula f^2 = (1/2)(a^2 + b^2 - c^2/2).

Interactive GEOMETOR construction for F.46 (Stewart’s Theorem). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point D : Point ===== a = segment(B, C) b = segment(C, A) c = segment(A, B) f_cev = segment(C, D) H_C = foot(C, line(A, B))

Mathematical Synthesis

Dated November 18, 2013 (SeeligNotebooks), Seelig’s two-page proof of Stewart's theorem (“:term:`Stewart’s Theorem` (also called :person:`Apollonius`’ Theorem)”, after Stewart’s 1746 Some General Theorems of Considerable Use in the Higher Parts of Mathematics, Stewart1746) normalizes the base segments of \(c = AB\) by dimensionless barycentric weights \(k' = |AD|/c\) and \(k = |DB|/c\) (\(k + k' = 1\)). Applying the Law of Cosines (or Euclid II.12/II.13 via the altitude foot HC) to the supplementary angles at D, the cosine terms cancel upon weighting by \(k\) and \(k'\), leaving the universal cevian formula \(f^2 = k a^2 + k' b^2 - k k' c^2\).

On Folio 2, Seelig sets \(k = k' = 1/2\) to immediately recover Apollonius’s median length identity \(m_c^2 = \frac{1}{2}\left(a^2 + b^2 - \frac{1}{2}c^2\right)\). On the canonical \((5,6,7)\) triangle, this gives \(m_c^2 = \frac{1}{2}(25 + 36 - 49/2) = \frac{73}{4}\), so \(m_c = \frac{\sqrt{73}}{2}\).

Key Algebraic Identities

\[f^2 = k a^2 + k' b^2 - k k' c^2 \qquad (k + k' = 1,\; |AD| = k' c,\; |DB| = k c)\]

\[k = k' = \frac{1}{2} \implies m_c^2 = \frac{1}{2}\left(a^2 + b^2 - \frac{1}{2}c^2\right) = \frac{2 a^2 + 2 b^2 - c^2}{4}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.46 Stewart’s Theorem — Folios (Folios 0293–0294, 2 scanned manuscript pages in resources/seelig/46_stewarts-theorem)
Euclid’s Elements	II.12, II.13 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment, G.11 rectilineal figures
Encyclopedia of Triangle Centers	X(2) — Centroid (Kimberling1998)
Seelig Spreadsheets	Stewart's Theorem, Ceva & Menelaus, Centroid (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.6 Centroid, F.7 Ceva and Menelaus, F.35 Obtuse and Acute Triangles

Required for: F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety