Master Cross-Repository Concordance & SymPy Circle Audit

Mapping Casey’s Sequel (1888 / 1892) Across euclid, elements, altshiller-court, etc, pappus, and sympy.geometry

1. Cross-Repository Canon Concordance

Every proposition cluster in John Casey's Sequel to Euclid bridges classical Euclidean propositions (geometor/euclid, geometor/elements) forward to Nathan Altshiller-Court's College Geometry (geometor/altshiller-court), Clark Kimberling's Encyclopedia of Triangle Centers (geometor/etc), and projective configurations in geometor/pappus:

Master Concordance of Casey’s Sequel to Euclid (Units 1–6)

Unit	Casey Locus & Folio (pp.)	Theorem / Topic	euclid / elements	altshiller-court (1952)	etc Centers	geometor.casey Symbol
Unit 1	Book I, Prop. 10 Cor. · f. 0034 (p. 8)	Concurrency of Altitudes (Orthocentre $H$)	I.12, I.31, I.34	§§ 93–96 (Ch. III)	X(4)	construct_euler_nine_point
Unit 1	Book I, Props. 17–18 · f. 0038–0039 (pp. 12–13)	Centres of Mean Position (Weighted Barycentres)	I.10, VI.2	§§ 65–72 (Ch. III)	X(2)	barycentric_point
Unit 1	Book I, Misc. Ex. 42–46 · f. 0046 (p. 20)	Euler Line ($O, G, N, H$ with $HG = 2\,GO$)	I.29, IV.5, VI.4	§§ 101–103 (Ch. III)	X(2), X(3), X(4), X(5)	construct_euler_nine_point
Unit 1	Book II, Props. 1–2, 7, 10 · f. 0048–0052 (pp. 22–26)	Apollonius Median, Euler Identity & Stewart’s Theorem	II.4, II.7, II.12, II.13	§§ 68, 243, 307–309	X(2)	prove_casey_theorem_symbolic
Unit 1	Book IV, Prop. 5 · f. 0083–0084 (pp. 57–58)	Nine-Points Circle ($R_9 = R/2$, 9 concyclic points)	III.21, III.31, IV.5	§§ 104–113 (Ch. III)	X(5)	construct_euler_nine_point
Unit 2	Book III, Props. 20–22 · f. 0067–0068 (pp. 41–42)	Inverse Points, Power of a Point & Orthogonal Circles	I.47, III.35–37	§§ 358–376, 417–428 (Ch. VIII)	—	are_orthogonal_circles
Unit 2	Book III, Props. 23–24, 28 & VI.V · f. 0069–0070, 0139–0145	Radical Axis, Radical Centre, Coaxal Pencils & Limiting Points	III.17, III.36	§§ 429–498 (Ch. VIII)	—	construct_radical_axis_and_coaxial_pencil
Unit 3	Book VI, Sec. I, Props. 2 & 4 · f. 0093–0095 (pp. 67–69)	Ceva’s Concurrency & Menelaus’s Transversal Theorems	VI.1, VI.2	§§ 313–335 (Ch. VI)	X(1), X(2), X(4), X(7), X(8)	barycentric_point
Unit 3	Book VI, Sec. II, Props. 1–6 · f. 0108–0113 (pp. 82–87)	Centres of Similitude & Monge’s Four Axes Theorem	VI.3, VI.4	§§ 23–53, 402–416	X(55), X(56)	radical_axis_line
Unit 3	Book VI, Sec. III & Sec. VI · f. 0113–0120, 0153–0166	Harmonic Section, Complete Quadrilateral, Anharmonic Ratio, Desargues & Pascal	VI.3, VI.A	§§ 339–357 (Chs. VI–VII)	(Pappus VII.129–139)	construct_complete_quadrilateral_harmonic
Unit 4	Book VI, Sec. IV, Props. 1–9 & Cor. 1 · f. 0121–0129 (pp. 95–103)	Circular Inversion & Pappus’s Arbelos Chain	III.32, III.36, VI.6	§§ 518–535 (Ch. IX)	(Pappus IV)	invert_point
Unit 4	Book VI, Sec. IV, Props. 10–11 & Cor. 2 · f. 0129–0134 (pp. 103–108)	Casey’s Generalized Ptolemy Theorem, Feuerbach’s Theorem & Hart’s Circle	IV.4, VI.D	§§ 200–205, 502–504, 536–546	X(1), X(5), X(11)	construct_casey_feuerbach
Unit 4	Book III, Props. 25, 27 & VI.VII · f. 0070–0073, 0167–0173	Poles & Polars, La Hire’s Reciprocity, Self-Conjugate Triangles & Brianchon	III.17, III.36	§§ 377–401 (Ch. VIII)	(Dual of Pascal)	construct_pole_and_polar
Unit 5	Supp. Ch., Secs. I & III · f. 0191–0198, 0205–0209 (pp. 165–183)	Isogonal Conjugates, Symmedians, Lemoine’s Point $K$, Lemoine & Cosine Circles	III.32, VI.2	§§ 557–590, 616–628, 661–673	X(6), X(182)	construct_symmedian_and_lemoine_circles
Unit 6	Supp. Ch., Secs. II–VII & Misc. · f. 0201–0274 (pp. 175–248)	Brocard Points $\Omega, \Omega'$, Brocard Circle, Tucker/Taylor, Harmonic Polygons, Steiner, Tarry & Longchamps	III.21, III.32, VI.4, VI.18	§§ 629–689 (Ch. X)	X(3), X(6), X(20), X(39), X(98), X(99), X(182)	construct_brocard_and_lemoine_circles

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2. sympy.geometry.Circle Capability Audit vs. geometor.casey

Inspection of sympy.geometry.curve.Circle in SymPy confirms that standard sympy.geometry provides basic Euclidean primitives (center, radius, equation, tangent_lines, is_tangent) yet omits the entire modern Euclidean circle-algebra toolkit codified by Casey. geometor.casey supplies exact closed-form symbolic implementations for every gap (audited programmatically via audit_sympy_circle_gaps()):

Audit of sympy.geometry.Circle Capabilities vs. geometor.casey

Euclidean Circle Concept	In sympy.geometry?	geometor.casey Implementation	Algebraic Invariant / Upstream Candidate Note
Power of a Point $\operatorname{pow}(P, C)$	Missing (False)	circle_power()	Exact rational/radical $|PO|^2 - r^2$ (Casey III.20–24).
Circle Orthogonality ($C_1 \perp C_2$)	Missing (False)	are_orthogonal_circles()	Exact zero_test(|O1 O2|^2 - r1^2 - r2^2) (Casey III.22).
Radical Axis & Radical Centre	Missing (False)	radical_axis_line() / construct_radical_axis_and_coaxial_pencil()	Linear difference $C_1(x, y) - C_2(x, y) = 0$ + compass construction for disjoint circles (Casey III.21, 23–24).
Poncelet Limiting Points $L_1, L_2$	Missing (False)	construct_radical_axis_and_coaxial_pencil()	Real zero-radius members $D \pm |DT_1|\,\hat{u}$ of non-intersecting coaxal pencil (Casey III.23, VI.V).
Circular Inversion $P \mapsto P'$	Missing (False)	invert_point()	Exact conformal map $O + \frac{r^2}{|OP|^2}(P - O)$ (Casey VI.IV.1–9).
Pole, Polar & La Hire Reciprocity	Missing (False)	polar_line(), pole_of_line()	Bilinear polarization $(x_0 - h)(x - h) + (y_0 - k)(y - k) = r^2$ + pure-straightedge quadrangle construction (Casey III.25, VI.VII).
Direct & Transverse Bitangent Quadrance	Missing (False)	bitangent_quadrance(), bitangent_length()	Exact $|O_1 O_2|^2 - (r_1 \mp r_2)^2$ powering Casey’s Generalized Ptolemy Theorem (Casey VI.IV.9–10).