W.12 Vertex Bisectors, Square Vertices & The Four Incenters

Number-Theoretic Criterion for Rational Bisectors and Symmetric Inquadrances

The locus of points X having equal quadrance Q(X, l_1) = Q(X, l_2) to two intersecting non-null lines l_1, l_2 splits into two orthogonal vertex bisectors over F if and only if (a_1^2 + b_1^2)(a_2^2 + b_2^2) is a square in F; when two vertices of a triangle have bisectors in F, so does the third, producing four symmetric incenters I_0, I_1, I_2, I_3.

Interactive GEOMETOR construction for W.12 (Vertex Bisectors, Square Vertices & The Four Incenters). 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) I_0 = incenter(A_1, A_2, A_3) b_1 = line(A_1, I_0) b_2 = line(A_2, I_0) b_3 = line(A_3, I_0) c_in = circle(I_0, foot(I_0, line(A_1, A_2)))

Mathematical Synthesis

Unlike medians and altitudes—which always exist rationally over \(F\)—vertex bisector lines reveal a deep arithmetic obstruction in Universal Geometry (Wildberger2005). Equating \(Q(X, \ell_1) = Q(X, \ell_2)\) gives \((a_2^2 + b_2^2)(a_1 x + b_1 y + c_1)^2 = (a_1^2 + b_1^2)(a_2 x + b_2 y + c_2)^2\), which factors into two perpendicular linear equations over \(F\) if and only if the vertex is a square vertex (\((a_1^2 + b_1^2)(a_2^2 + b_2^2) = r^2\) in \(F\), equivalently \(1 - s\) is a square when one line has norm square).

When two vertices of t admit bisectors in \(F\), Theorem 88 guarantees that the third vertex also admits bisectors, intersecting in four completely symmetric incenter points \(I_0, I_1, I_2, I_3\) (classical \(X(1)\) I0 and the three excenters). For the Heronian triangle \(A_1=[0,0], A_2=[14,0], A_3=[5,12]\), all three vertices are square over \(\mathbb{Q}\), placing I0 at exact integer coordinates \([6, 4]\) with inquadrance \(R_0 = 16\) (cin)!

Key Algebraic Identities

\[Q(X, \ell_1) = Q(X, \ell_2) \iff (a_2^2 + b_2^2)(a_1 x + b_1 y + c_1)^2 = (a_1^2 + b_1^2)(a_2 x + b_2 y + c_2)^2\]

\[s(I A_1, I A_2) = 1 - s(A_3 I, A_3 A_1), \qquad \mathcal{A}(Q_1, Q_2, Q_3) \text{ & } R_i:\; Q(I_0, \ell_1) = Q(I_0, \ell_2) = Q(I_0, \ell_3) = R_0\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Square vertex norms (|l_2|^2*|l_3|^2, ...)	(169*196, 225*196, 169*225)	All three products are exact squares in Q!
Rational Incenter I_0 = X(1)	[6, 4]	Exact integer coordinates
Equal side quadrances Q(I_0, l_i) = R_0	16 == 16 == 16	Inquadrance R_0 = 16
Incenter Spread Theorem (Thm 89)	s(I_0 A_1, I_0 A_2) = 49/65	Matches 1 - s(A_3 I_0, A_3 A_1) = 49/65

Cross-Repository Canon

Executable Demonstration	W.12 Vertex Bisectors, Square Vertices & The Four Incenters — Code (Standalone model.py & w-12.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Euclid’s Elements	I.9, IV.4 (Angle bisection and inscribed circle, EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.8 boundary
GEOMETOR Fundamentals	F.28 Incenter, F.29 Inradius (SeeligNotebooks)
ETC Reference	Incenter X(1) & Excenters in geometor/etc (Kimberling1998)

Constructive Dependency Graph

Depends on: W.02 Lineations, Central Rotations & Reflections in Lines, W.03 Quadrance, Midpoints & Point-to-Line Separation, W.04 Spread, Cross, Twist & Line Separation Ratios, W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas