Unit 1 Summary: Triangle Concurrency, Mean Centres, Medians & the Nine-Points Circle

Mathematical Synthesis of Casey’s Books I, II & IV Extensions (pp. 1–63)

1. Architectural Overview

Unit 1 gathers John Casey's foundational extensions to Euclid’s Books I, II, and IV before circle-power and projective methods enter in Units 2–4:

  • Four Classical Triangle Concurrencies (Book I, Props. 9–11): Internal/external angle bisectors meet at the Incentre $I = X(1)$ and Excentres $I_a, I_b, I_c$; perpendicular bisectors meet at the Circumcentre $O = X(3)$; altitudes meet at the Orthocentre $H = X(4)$ (proved via the circumscribed anticomplementary triangle); medians meet at the Centroid $G = X(2)$ in ratio $2:1$.

  • Weighted Mean Centres & Variational Extremes (Book I, Props. 17–22): Casey introduces weighted barycentres $G = (\sum m_i P_i)/(\sum m_i)$ synthetically and proves Heron’s reflection principle and the cyclic maximum-area property for polygons of given sides.

  • Quadratic Segment Identities (Book II, Props. 1–2, 7, 10): Apollonius's median theorem, Euler's quadrilateral midpoint identity, Euler’s collinear four-point identity $AB\cdot CD + BC\cdot AD = AC\cdot BD$, and Stewart's theorem $b^2 m + c^2 n = a(d^2 + mn)$.

  • The Euler Line & Nine-Points Circle (Book I Ex. 43; Book IV, Prop. 5): Collinearity of $O, G, N, H$ with $HG = 2\,GO$ and $HN = NO$, and concyclicity of the nine remarkable points on $(N, R/2)$.

2. Canonical Heronian Ledger on $\triangle ABC$ $(15, 13, 14)$

Verified symbolically in construct_euler_nine_point() on $A = (0, 0), B = (14, 0), C = (5, 12)$:

Exact Rational Coordinates & Radii on $(15, 13, 14)$

Geometric Object

Kimberling / Casey Symbol

Exact Rational Value

Area $\Delta$ & Semiperimeter $s$

$\Delta, s$

$\Delta = 84, \; s = 21$

Circumcentre & Circumradius

$O = X(3), \; R$

$O = \left(7, \frac{33}{8}\right), \; R = \frac{65}{8}$

Centroid

$G = X(2)$

$G = \left(\frac{19}{3}, 4\right)$

Orthocentre

$H = X(4)$

$H = \left(5, \frac{15}{4}\right)$

Nine-Points Centre & Radius

$N = X(5), \; R_9$

$N = \left(6, \frac{63}{16}\right), \; R_9 = \frac{65}{16}$

Euler Line Equation

$[O H]$

$3x - 16y + 45 = 0$

Side Midpoints

$M_a, M_b, M_c$

$\left(\frac{19}{2}, 6\right), \; \left(\frac{5}{2}, 6\right), \; (7, 0)$

Altitude Feet

$D_a, D_b, D_c$

$\left(\frac{224}{25}, \frac{168}{25}\right), \; \left(\frac{350}{169}, \frac{840}{169}\right), \; (5, 0)$

Euler Midpoints

$E_a, E_b, E_c$

$\left(\frac{5}{2}, \frac{15}{8}\right), \; \left(\frac{19}{2}, \frac{15}{8}\right), \; \left(5, \frac{63}{8}\right)$