Unit 6 Notes: Brocard, Neuberg, Casey’s Harmonic Polygons & Kimberling’s ETC

International Collaboration Behind the 1888–1892 Supplementary Chapter

1. Casey’s Transnational Network (Brocard, Neuberg, Tucker, M’Clelland)

In the Preface to the Fifth Edition (f. 0008, p. vi), John Casey explicitly acknowledges his personal correspondence with Joseph Neuberg (Liège), Henri Brocard (Montpellier/Algiers), Robert Tucker (London), and W. J. M'Clelland (Dublin). Notice two historical landmarks unique to Unit 6:

• Brocard’s 1881 Algiers Memoir (:cite:`AFAS Alger, 1881 <brocard:1881>`): Although Crelle (1816) and Jacobi (1825) had briefly studied the equi-angle points $\Omega, \Omega'$, Brocard discovered the Seven-Point Brocard Circle on diameter $[OK]$ unifying $O, K, \Omega, \Omega'$ and $\triangle A_1 B_1 C_1$.

• Harmonic Polygons (Casey & Neuberg, 1885–1886): After Neuberg introduced harmonic quadrilaterals ($AB\cdot CD = BC\cdot DA$, Mathesis 5, 1885), Casey immediately generalized the entire Lemoine–Brocard–Tucker apparatus to arbitrary cyclic Harmonic $n$-gons in the Proceedings of the Royal Irish Academy (1886) and incorporated that memoir directly as Section VI of the Supplementary Chapter (pp. 205–218).

2. Cross-Canon Links (geometor/etc & geometor/altshiller-court)

Every special point in Unit 6 maps directly into geometor/etc (Kimberling's ETC) and geometor/altshiller-court (College Geometry, Ch. X, §§ 629–689):

• X(3) Circumcentre $O$ & X(6) Symmedian Point $K$ (diameter of Brocard Circle);

• X(20) de Longchamps Point $L = 2O - H$ (Casey Misc. Ex. 146–158, p. 247);

• X(39) Brocard Midpoint $\frac{1}{2}(\Omega + \Omega')$ on $[OK]$;

• X(98) Tarry Point $T_r$ & X(99) Steiner Point $S_t$ (antipodal pair on the circumcircle);

• X(182) Midpoint $M_L = \frac{1}{2}(O + K)$, common centre of Lemoine’s First Circle and the Brocard Circle;

• Bicentric pair $(\Omega, \Omega')$: note that because $\Omega$ and $\Omega'$ permute cyclically rather than symmetrically under swap of $B \leftrightarrow C$, they form the prototype bicentric pair (rather than a single symmetric Kimberling triangle center), while their midpoint X(39) and line ΩΩ' are fully symmetric!