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

Exact Collinearity of C, G, N, O and the K/4 Nine-Point Circle in Q and F_p

In any non-null triangle A_1 A_2 A_3 over a field F (char F != 2), the nine-point center N = (C + O)/2 satisfies 2*N - C = O (and 3*G = 2*C + O when char F != 3), placing C, G, N, O on the Euler line with quadrance proportions Q(C, G) : Q(C, N) : Q(C, O) = 4 : 9 : 36; the three side midpoints M_i, three altitude feet F_i, and three Euler midpoints E_i = (A_i + O)/2 all have exact quadrance K / 4 from N.

Interactive GEOMETOR construction for W.16 (The Euler Line & Nine-Point Circle Over Arbitrary Fields). 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) C = circumcenter(A_1, A_2, A_3) O = orthocenter(A_1, A_2, A_3) N = midpoint(C, O) G = centroid(A_1, A_2, A_3) l_euler = line(C, O) c_circ = circle(C, A_1) c_nine = circle(N, midpoint(A_2, A_3))

Mathematical Synthesis

Leonhard Euler’s collinearity of C (\(X(3)\)), G (\(X(2)\)), N (\(X(5)\)), and O (\(X(4)\)) along leuler is a pure vector identity \(O - C = (A_1 - C) + (A_2 - C) + (A_3 - C)\) that holds over every field of characteristic not \(2\) (Wildberger2005, CoxeterGreitzer1967). Notice that even in characteristic \(3\)—where the centroid G moves to infinity—the midpoint relation \(N = (C + O)/2\) and the nine-point circle remain completely intact!

The circumcircle ccirc has circumquadrance \(K\), while cnine centered at N passes through all nine canonical points—side midpoints M1, M2, M3, altitude feet F1, F2, F3, and Euler midpoints E1, E2, E3—with exact quadrance \(K_9 = K / 4\), verified identically over both \(\mathbb{Q}[x_i, y_i]\) and finite prime fields such as \(\mathbb{F}_{17}\)!

Key Algebraic Identities

\[O - C = \sum_{i=1}^{3} (A_i - C), \qquad 3 G = 2 C + O, \qquad N = \frac{C + O}{2}\]

\[Q(C, G) = \frac{1}{9} Q(C, O), \quad Q(C, N) = \frac{1}{4} Q(C, O), \quad Q(N, M_i) = Q(N, F_i) = Q(N, E_i) = \frac{K}{4}\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Euler collinearity det(C, G, O) & det(C, N, O)	0, 0	Verified == 0
Euler quadrance ratio Q(C,G):Q(C,N):Q(C,O)	4 : 9 : 36	Exact harmonic center spacing
All 9 points {M_i, F_i, E_i} to N	305/18	Equals K / 4 = 610/9 / 4
Nine-Point Circle over GF(17)	N = [14, 1], K_9 = 16 (mod 17)	Exact modular verification

Cross-Repository Canon

Executable Demonstration	W.16 The Euler Line & Nine-Point Circle Over Arbitrary Fields — Code (Standalone model.py & w-16.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Euclid’s Elements	III.31, IV.5 (EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.4 segment, G.7 plane surface
GEOMETOR Fundamentals	F.34 Nine-Point Circle (SeeligNotebooks)
ETC Reference	Centers X(2), X(3), X(4), X(5) in geometor/etc (Kimberling1998)

Constructive Dependency Graph

Depends on: W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants, W.13 Stewart’s Theorem, Median Quadrances & The Centroid, W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics

Required for: W.23 Chromogeometry & Universal Symmetric Bilinear Forms