Heron’s Formula Synthetic Proof & Quadrea (Metrica I.8)

Exact Compass-and-Straightedge Model of Hero’s Incircle & Cyclic Quadrilateral Proof — Codex Reference: Metrica Book I (Preface & Props. 1–9): Triangles, Heron’s Formula & √720

Interactive Construction (heron-i8-incircle.svg)

Hover or click any geometric label below—such as ABC, incenter H, cyclic quadrilateral CHBL, or extended semiperimeter CG—to trace Hero's synthetic proof step-by-step:

Hero Metrica I.8 — Synthetic Incircle & Cyclic Quadrilateral Proof on (13, 14, 15) ( H E ): (x - 6)**2 + (y - 4)**2 - 16 = 0 ( M C ): (x - 7)**2 + (y + 4)**2 - 65 = 0 [ B C ]: y = 0 [ A B ]: 12*x - 5*y = 0 [ C A ]: -12*x - 9*y + 168 = 0 [ B E ]: -12*x + 18*y = 0 [ C G ]: -56*x/5 - 112*y/5 + 784/5 = 0 [ A H ]: 8*x + y - 52 = 0 [ H J ]: x - 6 = 0 [ T U ]: -16*sqrt(3)*x + 8*sqrt(3)*y + 64*sqrt(3) = 0 [ W X ]: x = 0 [ C L ]: 8*x - 14*y - 112 = 0 / H D /: |seg| = 4 / H E /: |seg| = 4 / H F /: |seg| = 4 / B G /: |seg| = 7 / A D /: |seg| = 7 / A F /: |seg| = 7 / B D /: |seg| = 6 / B E /: |seg| = 6 / C E /: |seg| = 8 / C F /: |seg| = 8 / C G /: |seg| = 21 B = (0, 0) C = (14, 0) A = (5, 12) H = (6, 4) E = (6, 0) D = (30/13, 72/13) F = (46/5, 32/5) G = (-7, 0) K = (4, 0) L = (0, -8) M = (7, -4) B C A H E D F G K L M

Exact Symbolic Coordinates & Deductive Chain

On Hero’s synthesis triangle \((AB, BC, CA) = (13, 14, 15)\) placed at B \(= (0, 0)\), C \(= (14, 0)\), and A \(= (5, 12)\):

  1. Incenter & Inradius: Internal angle bisectors meet at H \(= (6, 4)\) (X(1) in Kimberling's ETC), with touchpoints E \(= (6, 0)\), D \(= \left(\frac{30}{13}, \frac{72}{13}\right)\), and F \(= \left(\frac{46}{5}, \frac{32}{5}\right)\) of radius HE \(r = 4\).

  2. Extended Semiperimeter: Compass extension past B locates G \(= (-7, 0)\) so that BG = AD \(= s - a = 7\) and CG \(= s = 21\), certifying \(\Delta = s \cdot r = 21 \times 4 = 84\).

  3. Cyclic Auxiliary Quadrilateral: Erecting \(HL \perp CH\) at H (meeting BC at K \(= (4, 0)\)) and \(BL \perp BC\) at B meets at L \(= (0, -8)\). Quadrilateral CHBL is inscribed in circle (MC) with center M \(= (7, -4)\) and squared radius \(Q(M, C) = 65\).

  4. Right-Triangle Altitude Identity (:just:`VI.8`): In right triangle \(KHC\), altitude HE satisfies \(|KE| \cdot |EC| = 2 \times 8 = 16 = 4^2 = r^2\), completing Hero’s proportion:

    \[\Delta^2 = |CG|^2 \cdot |HE|^2 = |CG| \cdot |BG| \cdot |BE| \cdot |EC| = 21 \times 7 \times 6 \times 8 = 7056 = 84^2.\]

  5. Wildberger’s Polynomial Quadrea: Over \(\mathbb{Z}[Q_a, Q_b, Q_c]\) with \((Q_a, Q_b, Q_c) = (196, 225, 169)\):

    \[\mathcal{A} = (196 + 225 + 169)^2 - 2(196^2 + 225^2 + 169^2) = 112{,}896 = 16 \times 7056.\]

Executable Python API

from geometor.heron import archimedes_quadrea, construct_heron_proof, verify_quadrea_identity

        model, inv = construct_heron_proof()
        assert inv["area"] == 84
        assert inv["quadrea"] == archimedes_quadrea(196, 225, 169) == 112896
        assert verify_quadrea_identity() == 0