W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics

Radical-Free Coordinates for X(3), X(4), and Orthic Bisector Symmetry

Every non-null triangle A_1 A_2 A_3 over a field F (char F != 2) has a unique circumcenter C at the intersection of its side perpendicular bisectors and a unique orthocenter O at the intersection of its altitudes, with affine barycentric weights beta_i = Q_i*(Q_j + Q_k - Q_i)/(2*A) for C and gamma_i = (Q_i + Q_j - Q_k)*(Q_k + Q_i - Q_j)/(2*A) for O.
t t_orth c_circ (Q = 4225/64) / A_1 F_1 / (Q = 3136/25) / A_2 F_2 / (Q = 28224/169) / A_3 F_3 / (Q = 144) A_1 = [0, 0] A_2 = [14, 0] A_3 = [5, 12] C = [7, 33/8] O = [5, 15/4] F_1 = [224/25, 168/25] F_2 = [350/169, 840/169] F_3 = [5, 0] C = [7, 33/8] O = [5, 15/4] A1A1 A2A2 A3A3 CC OO F1F1 F2F2 F3F3 + − ⟲ Lbl Gde

Interactive GEOMETOR construction for W.15 (Circumcenter, Orthocenter & Universal Affine Barycentrics). 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) c_circ = circle(C, A_1) F_1 = foot(A_1, line(A_2, A_3)) F_2 = foot(A_2, line(A_1, A_3)) F_3 = foot(A_3, line(A_1, A_2)) O = meet(line(A_1, F_1), line(A_2, F_2)) t_orth = polygon(F_1, F_2, F_3)

Mathematical Synthesis

Unlike the incenter—which requires square vertices—both the circumcenter C (\(X(3)\)) and the orthocenter O (\(X(4)\)) are universally rational in the vertex coordinates of any non-null triangle t (Wildberger2005). Their homogeneous barycentric weights depend exclusively on the Conway-style dot-product factors \(2 S_i = Q_j + Q_k - Q_i\) normalized by \(2\mathcal{A}\), linking directly to F.14 Conway Notation and F.52 Homogeneous Barycentric Coordinates.

Dropping altitudes to feet F1, F2, F3 forms the orthic triangle torth, whose altitudes bisect its vertex spreads (Orthic Triangle Theorem): the altitudes of t are vertex bisector lines of torth, making O an incenter/excenter of torth even over fields where t itself has no rational incenter! Moreover, reflecting O across each sideline lands directly on the circumcircle ccirc.

Key Algebraic Identities

\[C = \sum_{i=1}^{3} \frac{Q_i(Q_j + Q_k - Q_i)}{2\mathcal{A}}\, A_i, \qquad O = \sum_{i=1}^{3} \frac{(Q_i + Q_j - Q_k)(Q_k + Q_i - Q_j)}{2\mathcal{A}}\, A_i\]
\[Q(O, A_3) = 4 K (1 - s_3), \qquad s(F_3 F_1, F_3 A_3) = s(F_3 F_2, F_3 A_3) = 1 - s_3\]

Symbolic SymPy Verification

Invariant / Theorem Check

Exact SymPy Value

Verification Status

Circumcenter C = X(3)

[7, 33/8]

K = 4225/64

Orthocenter O = X(4)

[5, 15/4]

Concurrent altitudes

Orthocenter quadrance Q(O, A_3) = 4*K*(1 - s_3)

1089/16 == 1089/16

Exact rational identity

Orthic bisector law s(F_3 F_1, F_3 A_3) == s(F_3 F_2, F_3 A_3)

1089/4225 == 1089/4225 = 1 - s_3

Theorem 86 verified

Cross-Repository Canon

Executable Demonstration

W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics — Code (Standalone model.py & w-15.svg)

Thematic Part

Part III: Universal Triangle, Conic & Chromogeometry

Euclid’s Elements

IV.5 (Circumscribing a circle about a triangle, EuclidElements)

GEOMETOR Elements

G.3 extremities of a segment, G.4 segment, G.7 plane surface

GEOMETOR Fundamentals

F.11 Circumcenter, F.36 Orthic Triangle, F.52 Homogeneous Barycentric Coordinates

ETC Reference

Circumcenter X(3) & Orthocenter X(4) (Kimberling1998)

Constructive Dependency Graph

digraph dependency { rankdir=TB; bgcolor="transparent"; node [fontname="Helvetica,Arial,sans-serif", fontsize=10, shape=box, style="rounded,filled", margin="0.15,0.08"]; edge [color="#888888", arrowsize=0.7]; "W.01" [label="W.01\nPlanar Points, Line Proportions...", fillcolor="#d8dee9", color="#4c566a", URL="../w-01/index.html"]; "W.03" [label="W.03\nQuadrance, Midpoints & Point-to...", fillcolor="#d8dee9", color="#4c566a", URL="../w-03/index.html"]; "W.08" [label="W.08\nLaw III: The Spread Law & Exten...", fillcolor="#d8dee9", color="#4c566a", URL="../w-08/index.html"]; "W.09" [label="W.09\nLaw IV: The Cross Law & Quadrea...", fillcolor="#d8dee9", color="#4c566a", URL="../w-09/index.html"]; "W.15" [label="W.15\nCircumcenter, Orthocenter & Uni...", fillcolor="#ffff99", color="#d4a017", penwidth=2.0, URL="../w-15/index.html"]; "W.16" [label="W.16\nThe Euler Line & Nine-Point Cir...", fillcolor="#e5e9f0", color="#777777", URL="../w-16/index.html"]; "W.01" -> "W.15"; "W.03" -> "W.15"; "W.08" -> "W.15"; "W.09" -> "W.15"; "W.15" -> "W.16"; }

Depends on: W.01 Planar Points, Line Proportions & Incidence Determinants, W.03 Quadrance, Midpoints & Point-to-Line Separation, W.08 Law III: The Spread Law & Extended Circumquadrance Law, W.09 Law IV: The Cross Law & Quadrea-Spread Identity

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