W.13 Stewart’s Theorem, Median Quadrances & The Centroid

Exact Quadratic Cevian Laws, Apollonius’ Identity, and X(2) Concurrency

For any affine point D = (1 - lambda)*A_1 + lambda*A_2 on side A_1 A_2 of triangle A_1 A_2 A_3, the cevian quadrance satisfies Q(A_3, D) = (1 - lambda)*Q_2 + lambda*Q_1 - lambda*(1 - lambda)*Q_3; at lambda = 1/2, the median quadrance satisfies 4*Q(A_3, M_3) = 2*Q_1 + 2*Q_2 - Q_3, and the three medians concur at G = (A_1 + A_2 + A_3)/3.

Interactive GEOMETOR construction for W.13 (Stewart’s Theorem, Median Quadrances & The Centroid). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_3 : Point ===== t = polygon(A_1, A_2, A_3) M_1 = midpoint(A_2, A_3) M_2 = midpoint(A_1, A_3) M_3 = midpoint(A_1, A_2) G = meet(line(A_1, M_1), line(A_2, M_2)) t_med = polygon(M_1, M_2, M_3)

Mathematical Synthesis

Matthew Stewart’s 1746 cevian relation becomes completely algebraic in quadrances (Wildberger2005). Expressing any point \(D\) on \(A_1 A_2\) as the affine combination \((1 - \lambda)A_1 + \lambda A_2\) (with \(R_1 = \lambda^2 Q_3\) and \(R_2 = (1 - \lambda)^2 Q_3\)) gives \(Q(A_3, D)\) as a linear combination of \(Q_1, Q_2, Q_3\), or in pure quadrances without \(\lambda\) via Stewart's theorem.

Setting \(\lambda = 1/2\) gives Apollonius’ Median Quadrance Theorem \(4 M_3 = 2 Q_1 + 2 Q_2 - Q_3\) for the midpoints M1, M2, M3. Provided \(\operatorname{char} F \neq 3\), the three medians concur at the centroid G \(= \frac{1}{3}(A_1 + A_2 + A_3)\) (\(X(2)\)), dividing each median in quadrance proportion \(4 : 1\), while the medial triangle tmed has side quadrances \(Q_i / 4\) and quadrea \(\mathcal{A} / 16\).

Key Algebraic Identities

\[Q(A_3, D) = (1 - \lambda) Q_2 + \lambda Q_1 - \lambda(1 - \lambda) Q_3, \qquad 4 Q(A_3, M_3) = 2 Q_1 + 2 Q_2 - Q_3\]

\[G = \frac{A_1 + A_2 + A_3}{3}, \qquad Q(A_i, G) = \frac{4}{9} Q(A_i, M_i) = 4 Q(G, M_i)\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Median quadrance 4 * Q(A_3, M_3)	360	Matches 2*Q_1 + 2*Q_2 - Q_3 = 360
Centroid G = (A_1 + A_2 + A_3)/3	[5, 3]	Concurrent medians
Centroid 4:1 quadrance division	Q(A_3,G) = 40, Q(G,M_3) = 10	Verified Q(A_3,G) == 4 * Q(G,M_3)
Medial triangle quadrea A_med	2916	Equals A / 16 = 2916

Cross-Repository Canon

Executable Demonstration	W.13 Stewart’s Theorem, Median Quadrances & The Centroid — Code (Standalone model.py & w-13.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Euclid’s Elements	I.10, VI.2 (Thales’ midpoint parallel theorem, EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.5 surface
GEOMETOR Fundamentals	F.6 Centroid, F.46 Stewart’s Theorem (SeeligNotebooks)
ETC Reference	Centroid X(2) in geometor/etc (Kimberling1998)

Constructive Dependency Graph

Depends on: W.01 Planar Points, Line Proportions & Incidence Determinants, W.03 Quadrance, Midpoints & Point-to-Line Separation, W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea, W.09 Law IV: The Cross Law & Quadrea-Spread Identity

Required for: W.16 The Euler Line & Nine-Point Circle Over Arbitrary Fields