Cross-Treatise Concordance

Unifying Euler (1765), Feuerbach (1822), Kimberling’s ETC, Euclid, Ceva, Menelaus, and Altshiller-Court

Because Leonhard Euler (1765/1767) and Karl Wilhelm Feuerbach (1822) wrote prior to modern standardized notation—and crucially reused letters such as \(G\), \(O\), and \(r\) with different meanings—this concordance maps every center, circle, invariant, and theorem across the historical editions and the GEOMETOR ecosystem.

Notation Rosetta Stone

Historical vs. Modern Triangle-Center & Invariant Symbols

Geometric Entity	Euler (1765, E325)	Feuerbach (1822)	ETC / Modern	Canonical (15, 13, 14) Value
Incenter	\(G\) (§ 4, § 8)	\(S\) (§ 1)	X(1) (\(I\))	\((6,\, 4)\)
Centroid	\(F\) (§ 3, § 7)	\(G\) (§ 60)	X(2) (\(G\))	\(\left(\frac{19}{3},\, 4\right)\)
Circumcenter	\(H\) (§ 5, § 9)	\(K\) (§ 4, § 49)	X(3) (\(O\))	\(\left(7,\, \frac{33}{8}\right)\)
Orthocenter	\(E\) (§ 2, § 6)	\(O\) (§ 19)	X(4) (\(H\))	\(\left(5,\, \frac{15}{4}\right)\)
Nine-Point Center	Midpoint of \(EH\) (§ 25)	\(L\) (§ 54, 9. Satz)	X(5) (\(N\))	\(\left(6,\, \frac{63}{16}\right)\)
Feuerbach Point	—	Inner tangency (§ 57)	X(11) (\(F_e\))	\((6,\, 8)\)
Three Excenters	—	\(S', S'', S'''\) (§ 1)	\(J_a, J_b, J_c\)	\((21, 14),\; (-7, \frac{21}{2}),\; (8, -12)\)
Excircle Tangencies	—	Outer tangencies (§ 57)	\(Fe_a, Fe_b, Fe_c\)	\((\frac{2709}{289}, \frac{1792}{289}),\; (\frac{553}{233}, \frac{1344}{233}),\; (\frac{1672}{257}, \frac{-24}{257})\)
Symbol \(r\)	Side product \(abc\) (§ 10)!	Inradius \(\Delta / s\) (§ 2)	\(abc\) vs. \(r\)	\(r_{\text{Euler}} = 2730\) vs. \(r_{\text{inc}} = 4\)
Triangle Area	\(A\) (§ 6)	\(\Delta\) (§ 2)	\(\Delta\)	\(84\)
Circumradius / Nine-Point Radius	\(\frac{r}{4A}\) (§ 20: \(R = 4R_{\text{circ}}^2\))	\(R\) and \(\frac{1}{2}R\) (§ 26)	\(R,\; R_9 = \frac{1}{2}R\)	\(R = \frac{65}{8},\; R_9 = \frac{65}{16}\)
Orthic Inradius	—	\(\rho = 2R\cos\alpha\cos\beta\cos\gamma\) (§ 24)	\(\rho\)	\(\frac{99}{52}\)

Theorem & Proposition Concordance

Cross-Corpus Structural & Algebraic Concordance

Theorem / Locus	Euler (1765/1767)	Feuerbach (1822)	Sister Codices	Symbolic Identity
Cevian Concurrency of Classical Centers	§§ 2–5 (Tab. II Fig. 1–4)	§ 1, § 19, § 60, Anhang 1.–4. Satz	ceva (1678); euclid IV.4, IV.5	\(\frac{AF}{FB}\frac{BD}{DC}\frac{CE}{EA} = 1\)
Euler Line & Harmonic Cross-Ratio	§§ 16–18 (\(EH = \frac{3}{2}EF\))	§§ 54–55, § 60 (\(GL = \frac{1}{6}KO\))	menelaus; pappus; altshiller-court § 191	\((X_4, X_2; X_5, X_3) = -1\)
Euler’s Incenter–Circumcenter Theorem	§ 15 & § 18 VI	§ 49 (explicitly cites Euler 1765 & Fuß 1797)	altshiller-court § 186; wildberger	\(Q(X_3, X_1) = R^2 - 2Rr\)
Euler–Stewart Four-Center Law	§ 18 (p. 114)	§ 60 (p. 40)	wildberger	\(4GH^2 + 2EG^2 = 3EF^2 + 6FG^2\)
Inverse Cubic Reconstruction	§§ 19–24, § 34	Vorrede (pp. V–XII)	etc	\(z^3 - pz^2 + qz - r = 0\)
Tritangent Radii Symmetry	§ 8, § 10	§§ 2–5, § 28 (cites L’Huilier 1810)	altshiller-court §§ 144–152	\(r' + r'' + r''' - r = 4R\)
Orthic Perimeter & Area Mean	—	§ 19 (cites Fagnano 1775), §§ 61–65, Anhang 5.–8. Satz	euclid VI.4, VI.16	\(\mu : \delta = \delta : \Delta\)
Nine-Point Circle Locus	—	§§ 26, 54–56, Anhang 9. Satz (p. 62)	altshiller-court §§ 195–201; etc X(5)	\(LM = LN = LP = \frac{1}{2}R\)
Feuerbach’s Fourfold Tangency Theorem	—	§ 57 (pp. 37–38, Tab. II Fig. 7)	etc X(11); altshiller-court § 215	\(LS = |\frac{1}{2}R - r|,\; LS^{(k)} = \frac{1}{2}R + r^{(k)}\)