modules

geometor.casey

Exact symbolic geometry constructions and Digital Scriptorium apparatus for John Casey’s Sequel.

Models the landmark theorems and straightedge-and-compass constructions of John Casey’s A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry (5th ed. 1888 / 6th ed. 1892):

• Euler Line & Nine-Points Circle (construct_euler_nine_point())

• Radical Axis, Coaxal Circles & Poncelet Limiting Points (construct_radical_axis_and_coaxial_pencil(), verify_coaxial_fundamental_theorem())

• Complete Quadrilateral Harmonic Division (construct_complete_quadrilateral_harmonic())

• Casey’s Generalized Ptolemy Theorem & Feuerbach’s Theorem (prove_casey_theorem_symbolic(), prove_casey_feuerbach_heronian(), construct_casey_feuerbach())

• Poles, Polars, Harmonic Secants & La Hire’s Reciprocity (construct_pole_and_polar(), verify_lahire_reciprocity_symbolic())

• Symmedians, Lemoine Circles, Brocard Points, Brocard Circle, Steiner & Tarry Points (construct_symmedian_and_lemoine_circles(), construct_brocard_and_lemoine_circles())

Submodules

• geometor.casey.__main__
• geometor.casey.brocard
• geometor.casey.coaxial
• geometor.casey.concordance
• geometor.casey.feuerbach
• geometor.casey.invariants
• geometor.casey.poles_polars
• geometor.casey.render

Attributes

HERONIAN_VERTICES	Canonical Heronian (15, 13, 14) scalene triangle A=(0,0), B=(14,0), C=(5,12).
EMBEDDED_SVG_STYLE
__version__

Classes

BrocardLemoineReport	Exact rational invariant summary for Symmedian, Lemoine & Brocard geometry on (15,13,14).
BrocardLemoineResult	Constructed elements for Brocard Points, Brocard Circle, Lemoine Circle, Steiner & Tarry.
LemoineCirclesResult	Constructed elements for Symmedians, First Lemoine Circle & Cosine Circle (Unit 5).
CoaxialPencilResult	Constructed elements for the Disjoint Coaxal Pencil & Poncelet Limiting Points (Unit 2).
CoaxialTheoremReport	Symbolic verification certificate for Casey's Fundamental Theorem (Book III, Prop. 24).
ConcordanceEntry	Single proposition/topic concordance mapping across the GEOMETOR canon.
SympyCircleGap	Audit record comparing sympy.geometry.Circle against geometor.casey.
CaseyFeuerbachHeronianReport	Exact rational invariant report for Casey's Theorem & Feuerbach tangency on (15, 13, 14).
CaseyFeuerbachResult	Constructed elements for Casey's & Feuerbach's Theorem on (15, 13, 14) (Unit 4).
CaseySymbolicProof	Symbolic verification certificate for Casey's Generalized Ptolemy Theorem.
EulerNinePointResult	Constructed elements for the Euler Line and Nine-Points Circle (Unit 1).
CompleteQuadrilateralResult	Constructed elements for Casey's Complete Quadrilateral Harmonic Section (Unit 3).
LaHireSymbolicProof	Universal symbolic proof certificate for La Hire's Reciprocity & Harmonic Secants.
PolePolarResult	Constructed elements for Pole, Polar, Harmonic Secants & La Hire Reciprocity (Unit 4).

Functions

construct_brocard_and_lemoine_circles([model]) → BrocardLemoineResult	Construct the Brocard Points, Brocard Circle, First Lemoine Circle, Steiner & Tarry Points.
construct_symmedian_and_lemoine_circles([model]) → LemoineCirclesResult	Construct Lemoine's Point K, First Lemoine Circle & Cosine Circle (Unit 5).
verify_brocard_identities_symbolic() → dict[str, sympy.Expr]	Verify Casey's universal algebraic identities for cot(omega) and K's minimum property.
construct_radical_axis_and_coaxial_pencil([model]) → CoaxialPencilResult	Construct the radical axis, coaxal system, orthogonal pencil, and limiting points.
verify_coaxial_fundamental_theorem(c1, c2, c3_coaxial, ...) → CoaxialTheoremReport	Verify Casey Book III Props. 22–24 symbolically and on the constructed coaxal system.
audit_sympy_circle_gaps() → list[SympyCircleGap]	Inspect sympy.geometry.Circle for modern-circle-geometry primitives in Casey.
get_resources_dir() → pathlib.Path	Return the path to resources/sequel_to_euclid/ inside the repository.
load_apparatus_registry(name) → dict[str, Any]	Load a JSON apparatus file (persons, citations, glossary, sources_manifest).
load_concordance() → list[ConcordanceEntry]	Load all cross-repository concordance entries from concordance.json.
compass_internal_angle_bisector(model, vertex, arm_p1, ...) → sympy.geometry.Line	Construct the internal bisector of < arm_p1 vertex arm_p2 ) by Euclid I.9.
construct_casey_feuerbach([model]) → CaseyFeuerbachResult	Construct Casey's Inversion Proof & Feuerbach's Tangency Theorem on (15, 13, 14).
construct_euler_nine_point([model]) → EulerNinePointResult	Construct the Euler Line and Nine-Points Circle by straightedge and compass (Unit 1).
prove_casey_feuerbach_heronian() → CaseyFeuerbachHeronianReport	Verify Casey's bitangent equation and all 4 Feuerbach tangencies on Heronian (15, 13, 14).
prove_casey_theorem_symbolic() → CaseySymbolicProof	Prove Casey's Generalized Ptolemy Theorem & Feuerbach's Corollary symbolically.
are_orthogonal_circles(c1, c2) → bool	Return True iff circles c1 and c2 cut orthogonally (Casey Book III, Prop. 22).
barycentric_point(a_pt, b_pt, c_pt, w_a, w_b, w_c) → sympy.geometry.Point	Compute the exact homogeneous barycentric point (w_a : w_b : w_c) on ABC.
bitangent_length(c1, c2, *[, internal]) → sympy.Expr	Compute the exact common-tangent length t_12 between two circles or points.
bitangent_quadrance(c1, c2, *[, internal]) → sympy.Expr	Compute the exact squared common-tangent length t_12^2 between circles or null-circles.
circle_power(pt, circ) → sympy.Expr	Compute the exact power |PO|^2 - r^2 of point pt with respect to circ.
compass_midpoint(model, p1, p2, *[, ID, classes]) → sympy.geometry.Point	Construct the exact midpoint of p1 p2 by straightedge and compass (Euclid I.10).
compass_parallel_line(model, line_p1, line_p2, through, *) → sympy.geometry.Line	Construct the line through through parallel to line_p1 line_p2 (Euclid I.31).
compass_perp_bisector(model, p1, p2, *[, ID, classes, ...]) → sympy.geometry.Line	Construct the perpendicular bisector of p1 p2 by Euclid I.10 symmetric circles.
compass_perpendicular_line(model, line_p1, line_p2, ...) → sympy.geometry.Line	Construct the perpendicular to line line_p1 line_p2 through through.
invert_point(pt, center, radius_sq) → sympy.geometry.Point	Compute the exact circular inverse of pt with respect to (center, sqrt(radius_sq)).
polar_line(pt, circ) → sympy.geometry.Line	Compute the exact polar line of pt with respect to circ (Casey Book III, Def. p. 44).
pole_of_line(line, circ) → sympy.geometry.Point	Compute the exact pole of line with respect to circ (Casey Book III, Def. p. 44).
radical_axis_line(c1, c2) → sympy.geometry.Line	Compute the exact radical axis of two non-concentric circles (Casey Book III, Prop. 23).
signed_triangle_area(a_pt, b_pt, c_pt) → sympy.Expr	Compute the exact signed area of oriented triangle ABC.
construct_complete_quadrilateral_harmonic([model]) → CompleteQuadrilateralResult	Construct Casey's Complete Quadrilateral and verify harmonic diagonal division (Unit 3).
construct_pole_and_polar([model]) → PolePolarResult	Construct the Polar of D by straightedge and verify La Hire's Reciprocity (Unit 4).
verify_lahire_reciprocity_symbolic() → LaHireSymbolicProof	Prove La Hire's Reciprocity (Book III, Prop. 25) and Harmonic Secant Section (VI.III.4).
export_svg(model, output_path, *[, graphic_id, title, ...]) → pathlib.Path	Write an interactive .euclid-graphic SVG rendering of model to output_path.
render_svg_string(model, *[, graphic_id, title, ...]) → str	Render a Model into a complete .euclid-graphic SVG string.

Package Contents

class geometor.casey.BrocardLemoineReport

Exact rational invariant summary for Symmedian, Lemoine & Brocard geometry on (15,13,14).

cot_omega: sympy.Rational

sin_sq_omega: sympy.Rational

symmedian_point: sympy.geometry.Point

brocard_center: sympy.geometry.Point

brocard_1: sympy.geometry.Point

brocard_2: sympy.geometry.Point

brocard_first_triangle: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

steiner_point: sympy.geometry.Point

tarry_point: sympy.geometry.Point

longchamps_point: sympy.geometry.Point

lemoine_intercept_ratios: tuple[sympy.Rational, sympy.Rational]

min_perp_squares_sum: sympy.Rational

verified: bool

class geometor.casey.BrocardLemoineResult

Constructed elements for Brocard Points, Brocard Circle, Lemoine Circle, Steiner & Tarry.

model: geometor.model.Model

vertices: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

circumcenter: sympy.geometry.Point

circumcircle: sympy.geometry.Circle

symmedian_point: sympy.geometry.Point

brocard_center: sympy.geometry.Point

brocard_1: sympy.geometry.Point

brocard_2: sympy.geometry.Point

brocard_first_triangle: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

brocard_circle: sympy.geometry.Circle

lemoine_hexagon: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

first_lemoine_circle: sympy.geometry.Circle

steiner_point: sympy.geometry.Point

tarry_point: sympy.geometry.Point

longchamps_point: sympy.geometry.Point

report: BrocardLemoineReport

class geometor.casey.LemoineCirclesResult

Constructed elements for Symmedians, First Lemoine Circle & Cosine Circle (Unit 5).

model: geometor.model.Model

vertices: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

circumcenter: sympy.geometry.Point

circumcircle: sympy.geometry.Circle

tangential_vertices: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

symmedian_point: sympy.geometry.Point

lemoine_center: sympy.geometry.Point

lemoine_hexagon: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

first_lemoine_circle: sympy.geometry.Circle

cosine_hexagon: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

cosine_circle: sympy.geometry.Circle

geometor.casey.construct_brocard_and_lemoine_circles(model: geometor.model.Model | None = None) → BrocardLemoineResult

Construct the Brocard Points, Brocard Circle, First Lemoine Circle, Steiner & Tarry Points.

On the Heronian (15, 13, 14) triangle A=(0,0), B=(14,0), C=(5,12):

1. Constructs O = (7, 33/8), K = X(6) = (1673/295, 1176/295), and the First Lemoine Circle centered at Ml = midpoint(O, K) = (4053/590, 9567/2360).

2. Constructs the First Brocard Point ``Om1`` via tangent circles (Ob1, B) (tangent to BC at B through A) and (Oc1, C) (tangent to CA at C through B), verifying concurrency with (Oa1, A) (tangent to AB at A through C).

3. Constructs the Second Brocard Point ``Om2`` via reversed tangent circles.

4. Constructs Brocard’s First Triangle ``A1 B1 C1`` (A1 = [B Om1] ∩ [C Om2], etc.) and proves that all seven points ``O, K, Om1, Om2, A1, B1, C1`` lie on the Brocard Circle with diameter OK (Casey, Supp. Sec. V, pp. 194–195).

5. Constructs Steiner’s Point ``St = X(99)`` (parallels through A, B, C to sides of A1 B1 C1), Tarry’s Point ``Tr = X(98)`` (antipode of St on circumcircle), and de Longchamps’ Point ``Lo = X(20)`` (reflection of H across O).

Parameters:

model – Optional existing Model.

Returns:

Populated BrocardLemoineResult with full exact symbolic report.

geometor.casey.construct_symmedian_and_lemoine_circles(model: geometor.model.Model | None = None) → LemoineCirclesResult

Construct Lemoine’s Point K, First Lemoine Circle & Cosine Circle (Unit 5).

Follows Casey, Supplementary Chapter, Sec. I, Props. 5–6 (pp. 170–171) and Sec. III, Prop. 1 & Cor. 4 (pp. 179–183) on A=(0,0), B=(14,0), C=(5,12).

Parameters:

model – Optional existing Model.

Returns:

Populated LemoineCirclesResult.

geometor.casey.verify_brocard_identities_symbolic() → dict[str, sympy.Expr]

Verify Casey’s universal algebraic identities for cot(omega) and K’s minimum property.

1. By the Law of Cosines, cot A + cot B + cot C = (b^2+c^2-a^2)/(4*Delta) + (c^2+a^2-b^2)/(4*Delta) + (a^2+b^2-c^2)/(4*Delta) = (a^2+b^2+c^2)/(4*Delta).

2. By Lagrange’s identity (Sec. I, Prop. 7, p. 171): (a^2+b^2+c^2)*(x^2+y^2+z^2) - (a*x+b*y+c*z)^2 = (b*z - c*y)^2 + (c*x - a*z)^2 + (a*y - b*x)^2, proving that under a*x+b*y+c*z = 2*Delta, x^2 + y^2 + z^2 attains its unique minimum when x/a = y/b = z/c (at K = X(6)).

Returns:

Dictionary of zero-verified symbolic residuals.

class geometor.casey.CoaxialPencilResult

Constructed elements for the Disjoint Coaxal Pencil & Poncelet Limiting Points (Unit 2).

model: geometor.model.Model

c1: sympy.geometry.Circle

c2: sympy.geometry.Circle

c3_coaxial: sympy.geometry.Circle

aux_secant_circle: sympy.geometry.Circle

radical_center: sympy.geometry.Point

radical_foot: sympy.geometry.Point

radical_axis: sympy.geometry.Line

line_of_centers: sympy.geometry.Line

tangent_contact_c1: sympy.geometry.Point

limiting_points: tuple[sympy.geometry.Point, sympy.geometry.Point]

orthogonal_circle_1: sympy.geometry.Circle

orthogonal_circle_2: sympy.geometry.Circle

report: CoaxialTheoremReport

class geometor.casey.CoaxialTheoremReport

Symbolic verification certificate for Casey’s Fundamental Theorem (Book III, Prop. 24).

symbolic_prop24_residual: sympy.Expr

radical_axis_power_diff: sympy.Expr

limiting_point_power_c1: sympy.Expr

limiting_point_power_c2: sympy.Expr

orthogonal_to_c1: bool

orthogonal_to_c2: bool

orthogonal_to_c3: bool

verified: bool

geometor.casey.construct_radical_axis_and_coaxial_pencil(model: geometor.model.Model | None = None) → CoaxialPencilResult

Construct the radical axis, coaxal system, orthogonal pencil, and limiting points.

Uses exact integer-rational circles C1 = ((-10, 0), 6) and C2 = ((17, 0), 15):

1. Draws an auxiliary secant circle C_aux = ((2, 8), 10) cutting C1 in U1, V1 and C2 in U2, V2 (Casey Book III, Prop. 28).

2. Intersects common chords [U1 V1] and [U2 V2] to obtain the Radical Centre Rc = (0, -6) (Book III, Prop. 23, Cor. 2).

3. Drops the perpendicular from Rc to the line of centres [O1 O2] to construct the Radical Axis x = 0 and foot D = (0, 0).

4. Constructs the tangent D T1 to C1 by Euclid III.17 (Thales circle on O1 D) and transfers |D T1| = 8 onto [O1 O2] via circle (D, T1) to construct Poncelet’s Limiting Points L1 = (-8, 0) and L2 = (8, 0).

5. Constructs two members (Rc, L1) and (D, T1) of the conjugate orthogonal pencil passing through L1, L2 and cutting C1, C2, C3 orthogonally.

Parameters:

model – Optional existing Model.

Returns:

Populated CoaxialPencilResult with exact coordinates and verification report.

geometor.casey.verify_coaxial_fundamental_theorem(c1: sympy.geometry.Circle, c2: sympy.geometry.Circle, c3_coaxial: sympy.geometry.Circle, orth_circle: sympy.geometry.Circle, limiting_points: tuple[sympy.geometry.Point, sympy.geometry.Point]) → CoaxialTheoremReport

Verify Casey Book III Props. 22–24 symbolically and on the constructed coaxal system.

Parameters:

• c1 – First primary circle of the coaxal pencil.

• c2 – Second primary circle of the coaxal pencil.

• c3_coaxial – Third circle belonging to the same coaxal pencil.

• orth_circle – Conjugate circle cutting the pencil orthogonally.

• limiting_points – Pair (L1, L2) of Poncelet limiting points.

Returns:

Frozen CoaxialTheoremReport certificate.

class geometor.casey.ConcordanceEntry

Single proposition/topic concordance mapping across the GEOMETOR canon.

id: str

unit: str

casey_locus: str

folio: str

page: str

title: str

euclid_refs: tuple[str, ...]

altshiller_court_refs: tuple[str, ...]

etc_centers: tuple[str, ...]

pappus_refs: tuple[str, ...]

casey_symbol: str

class geometor.casey.SympyCircleGap

Audit record comparing sympy.geometry.Circle against geometor.casey.

capability: str

sympy_method_present: bool

casey_locus: str

geometor_casey_api: str

geometor.casey.audit_sympy_circle_gaps() → list[SympyCircleGap]

Inspect sympy.geometry.Circle for modern-circle-geometry primitives in Casey.

Returns:

List of SympyCircleGap entries documenting native presence vs. geometor.casey exact symbolic implementations.

geometor.casey.get_resources_dir() → pathlib.Path

Return the path to resources/sequel_to_euclid/ inside the repository.

geometor.casey.load_apparatus_registry(name: str) → dict[str, Any]

Load a JSON apparatus file (persons, citations, glossary, sources_manifest).

Parameters:

name – Filename stem without .json extension.

Returns:

Parsed JSON dictionary.

geometor.casey.load_concordance() → list[ConcordanceEntry]

Load all cross-repository concordance entries from concordance.json.

Returns:

List of ConcordanceEntry records in codex order.

class geometor.casey.CaseyFeuerbachHeronianReport

Exact rational invariant report for Casey’s Theorem & Feuerbach tangency on (15, 13, 14).

sides: tuple[sympy.Integer, sympy.Integer, sympy.Integer]

area: sympy.Rational

inradius: sympy.Rational

circumradius: sympy.Rational

nine_point_radius: sympy.Rational

exradii: tuple[sympy.Rational, sympy.Rational, sympy.Rational]

midpoint_chords: tuple[sympy.Rational, sympy.Rational, sympy.Rational]

incircle_midpoint_tangents: tuple[sympy.Rational, sympy.Rational, sympy.Rational]

casey_signed_sum: sympy.Expr

feuerbach_point: sympy.geometry.Point

excircle_feuerbach_points: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

ni_distance: sympy.Rational

inversion_circle_orthogonal_to_incircle: bool

inversion_circle_orthogonal_to_excircle_a: bool

verified: bool

class geometor.casey.CaseyFeuerbachResult

Constructed elements for Casey’s & Feuerbach’s Theorem on (15, 13, 14) (Unit 4).

model: geometor.model.Model

euler_nine_point: EulerNinePointResult

incenter: sympy.geometry.Point

incircle_contacts: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

incircle: sympy.geometry.Circle

feuerbach_point: sympy.geometry.Point

casey_inversion_circle: sympy.geometry.Circle

fourth_common_tangent: sympy.geometry.Line

class geometor.casey.CaseySymbolicProof

Symbolic verification certificate for Casey’s Generalized Ptolemy Theorem.

inversion_invariance_residual: sympy.Expr

tangent_line_euler_residual: sympy.Expr

incircle_casey_residual: sympy.Expr

incircle_cor2_squares_residual: sympy.Expr

excircle_a_casey_residual: sympy.Expr

verified: bool

class geometor.casey.EulerNinePointResult

Constructed elements for the Euler Line and Nine-Points Circle (Unit 1).

model: geometor.model.Model

vertices: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

midpoints: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

altitude_feet: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

euler_midpoints: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

centroid: sympy.geometry.Point

circumcenter: sympy.geometry.Point

orthocenter: sympy.geometry.Point

nine_point_center: sympy.geometry.Point

euler_line: sympy.geometry.Line

circumcircle: sympy.geometry.Circle

nine_point_circle: sympy.geometry.Circle

geometor.casey.compass_internal_angle_bisector(model: geometor.model.Model, vertex: sympy.geometry.Point, arm_p1: sympy.geometry.Point, arm_p2: sympy.geometry.Point, *, ID: str = '', classes: list[str] | None = None, guide: bool = False) → sympy.geometry.Line

Construct the internal bisector of < arm_p1 vertex arm_p2 ) by Euclid I.9.

Transfers |vertex arm_p1| onto ray vertex arm_p2 via circle (vertex, arm_p1) and bisects the resulting isosceles base chord.

Parameters:

• model – Target Model.

• vertex – Angle vertex.

• arm_p1 – Point on the first ray.

• arm_p2 – Point on the second ray.

• ID – Optional explicit line identifier.

• classes – Optional semantic CSS classes.

• guide – Whether the angle bisector should remain a guide.

Returns:

Exact internal angle bisector sympy.geometry.Line.

geometor.casey.construct_casey_feuerbach(model: geometor.model.Model | None = None) → CaseyFeuerbachResult

Construct Casey’s Inversion Proof & Feuerbach’s Tangency Theorem on (15, 13, 14).

Builds the Nine-Points Circle, constructs the Incentre I and Incircle (I, Ta) by pure compass-and-straightedge (Euclid IV.4), erects Casey’s 1861 orthogonal inversion circle (Ma, Ta) (Book VI, Sec. IV, p. 106), constructs the fourth common internal tangent of (I, r) and (Ia, r_a) via reflection across line of centres AI, and discovers the exact Feuerbach point Fe = X(11) = (6, 8) as (N, Ma) ∩ (I, Ta).

Parameters:

model – Optional existing Model.

Returns:

Populated CaseyFeuerbachResult.

geometor.casey.construct_euler_nine_point(model: geometor.model.Model | None = None) → EulerNinePointResult

Construct the Euler Line and Nine-Points Circle by straightedge and compass (Unit 1).

Follows Casey, Book I, Misc. Ex. 42–46 (p. 20) and Book IV, Prop. 5 (pp. 57–58) on A=(0,0), B=(14,0), C=(5,12), verifying exact incidence of all nine classical points and Euler trisection |HG|^2 = 4*|GO|^2.

Parameters:

model – Optional existing Model. Created with auto_intersect=False when None.

Returns:

Populated EulerNinePointResult.

geometor.casey.prove_casey_feuerbach_heronian() → CaseyFeuerbachHeronianReport

Verify Casey’s bitangent equation and all 4 Feuerbach tangencies on Heronian (15, 13, 14).

Evaluates exact coordinates on A=(0,0), B=(14,0), C=(5,12) (a=15, b=13, c=14), confirming:

• Midpoint chords t_23 = 15/2, t_31 = 13/2, t_12 = 7.

• Midpoint-to-incircle bitangents |Ma Ta| = 1/2, |Mb Tb| = 1/2, |Mc Tc| = 1.

• Casey’s equation (15/2)*(1/2) + (13/2)*(1/2) - 7*1 = 15/4 + 13/4 - 7 = 0.

• Internal tangency of Nine-Points Circle N=(6, 63/16), R_9=65/16 with Incircle I=(6, 4), r=4 at Fe = X(11) = (6, 8) with |NI| = 1/16 = R_9 - r.

• External tangency of Nine-Points Circle with all three Excircles (Ia, r_a), (Ib, r_b), (Ic, r_c) with |N I_k| = R_9 + r_k.

• Casey’s 1861 inversion circle (Ma, 1/2) cuts both (I, 4) and (Ia, 14) orthogonally and inverts their fourth common internal tangent into the Nine-Points Circle.

Returns:

Frozen CaseyFeuerbachHeronianReport with exact rational values.

geometor.casey.prove_casey_theorem_symbolic() → CaseySymbolicProof

Prove Casey’s Generalized Ptolemy Theorem & Feuerbach’s Corollary symbolically.

Verifies four algebraic pillars from Casey, Book VI, Sec. IV, Props. 9–10 (pp. 102–106):

1. Inversion Invariance (Prop. 9, pp. 102–103): Under inversion in (O, k) at the origin, for two circles (x1, y1, r1) and (x2, y2, r2) not passing through O, the normalized bitangent ratio t_12^2 / (r1 * r2) is strictly invariant: t'_12^2 / (r'_1 * r'_2) - t_12^2 / (r1 * r2) == 0.

2. Collinear Contact Reduction (Prop. 10, pp. 103–104): Inverting from a point on the fifth tangent circle maps it to y = 0 and the four tangent circles to (u_i, r_i) with t_ij = |u_i - u_j|, reducing t_12*t_34 + t_14*t_23 = t_13*t_24 for ordered contacts u_1 < u_2 < u_3 < u_4 to Euler’s identity (Book II, Prop. 7, p. 24).

3. Universal Incircle Bitangent Identity (Prop. 10, Cor. 2, pp. 104–105): For general positive side lengths b <= c <= a, t_23 = a/2, t_31 = b/2, t_12 = c/2 and t_14 = (c - b)/2, t_24 = (a - c)/2, t_34 = (a - b)/2 satisfy t_23*t_14 + t_31*t_24 - t_12*t_34 == 0, alongside Casey’s squared identity (b^2 - c^2) + (c^2 - a^2) + (a^2 - b^2) == 0.

4. Universal Excircle Bitangent Identity: For the A-excircle, t_14 = (c - b)/2, t_24 = (a + c)/2, t_34 = (a + b)/2 satisfy t_23*t_14 + t_31*t_24 - t_12*t_34 == 0 (with t_31*t_24 = t_12*t_34 + t_23*t_14).

Returns:

Frozen CaseySymbolicProof certificate with zero residuals.

geometor.casey.HERONIAN_VERTICES: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

Canonical Heronian (15, 13, 14) scalene triangle A=(0,0), B=(14,0), C=(5,12).

geometor.casey.are_orthogonal_circles(c1: sympy.geometry.Circle, c2: sympy.geometry.Circle) → bool

Return True iff circles c1 and c2 cut orthogonally (Casey Book III, Prop. 22).

Two circles cut orthogonally iff the square of the distance between their centres equals the sum of the squares of their radii: |O1 O2|^2 = r1^2 + r2^2.

Parameters:

• c1 – First circle (O1, r1).

• c2 – Second circle (O2, r2).

Returns:

True if |O1 O2|^2 - r1^2 - r2^2 == 0 symbolically.

geometor.casey.barycentric_point(a_pt: sympy.geometry.Point, b_pt: sympy.geometry.Point, c_pt: sympy.geometry.Point, w_a: sympy.Expr, w_b: sympy.Expr, w_c: sympy.Expr) → sympy.geometry.Point

Compute the exact homogeneous barycentric point (w_a : w_b : w_c) on ABC.

Implements Casey’s Centre of Mean Position (Book I, Props. 17–18, pp. 12–13) and symmedian/Brocard weights (Supplementary Chapter, Secs. I–II).

Parameters:

• a_pt – First vertex A.

• b_pt – Second vertex B.

• c_pt – Third vertex C.

• w_a – Weight attached to A.

• w_b – Weight attached to B.

• w_c – Weight attached to C.

Returns:

Exact sympy.geometry.Point (w_a*A + w_b*B + w_c*C) / (w_a + w_b + w_c).

Raises:

ValueError – If w_a + w_b + w_c == 0.

geometor.casey.bitangent_length(c1: sympy.geometry.Circle | sympy.geometry.Point, c2: sympy.geometry.Circle | sympy.geometry.Point, *, internal: bool = False) → sympy.Expr

Compute the exact common-tangent length t_12 between two circles or points.

Parameters:

• c1 – First circle or null-circle point.

• c2 – Second circle or null-circle point.

• internal – If True, return the internal common-tangent length; else external.

Returns:

Exact symbolic principal square root sqrt(bitangent_quadrance(c1, c2)).

geometor.casey.bitangent_quadrance(c1: sympy.geometry.Circle | sympy.geometry.Point, c2: sympy.geometry.Circle | sympy.geometry.Point, *, internal: bool = False) → sympy.Expr

Compute the exact squared common-tangent length t_12^2 between circles or null-circles.

In Casey’s Generalized Ptolemy Theorem (Book VI, Sec. IV, Prop. 10, pp. 103–104), a point is treated as a circle of radius 0. For two circles soaked in the same sense (external common tangent), t_12^2 = |O1 O2|^2 - (r1 - r2)^2; for transverse (internal) contact, t_12^2 = |O1 O2|^2 - (r1 + r2)^2.

Parameters:

• c1 – First sympy.geometry.Circle or null-circle sympy.geometry.Point.

• c2 – Second sympy.geometry.Circle or null-circle sympy.geometry.Point.

• internal – If True, use (r1 + r2)^2 (internal bitangent); otherwise (r1 - r2)^2.

Returns:

Exact symbolic squared bitangent length.

geometor.casey.circle_power(pt: sympy.geometry.Point, circ: sympy.geometry.Circle) → sympy.Expr

Compute the exact power |PO|^2 - r^2 of point pt with respect to circ.

Corresponds to Casey, Book III, Props. 20–24 (pp. 41–44): for exterior points, equals the square of the tangent from pt to circ.

Parameters:

• pt – Query point P.

• circ – Reference circle (O, r).

Returns:

Exact symbolic expression (P.x - O.x)^2 + (P.y - O.y)^2 - r^2.

geometor.casey.compass_midpoint(model: geometor.model.Model, p1: sympy.geometry.Point, p2: sympy.geometry.Point, *, ID: str = '', classes: list[str] | None = None) → sympy.geometry.Point

Construct the exact midpoint of p1 p2 by straightedge and compass (Euclid I.10).

Parameters:

• model – Target Model.

• p1 – First endpoint.

• p2 – Second endpoint.

• ID – Optional explicit point identifier.

• classes – Optional semantic CSS classes.

Returns:

Exact midpoint sympy.geometry.Point.

geometor.casey.compass_parallel_line(model: geometor.model.Model, line_p1: sympy.geometry.Point, line_p2: sympy.geometry.Point, through: sympy.geometry.Point, *, ID: str = '', classes: list[str] | None = None, guide: bool = False) → sympy.geometry.Line

Construct the line through through parallel to line_p1 line_p2 (Euclid I.31).

Constructs the fourth vertex Q = line_p2 + through - line_p1 of parallelogram line_p1 line_p2 Q through by bisecting diagonal line_p2 through at M and doubling line_p1 M via circle (M, line_p1).

Parameters:

• model – Target Model.

• line_p1 – First point on the reference line.

• line_p2 – Second point on the reference line.

• through – Exterior point through which the parallel is drawn.

• ID – Optional explicit line identifier.

• classes – Optional semantic CSS classes.

• guide – Whether the parallel line should remain a guide.

Returns:

Exact parallel sympy.geometry.Line passing through through.

geometor.casey.compass_perp_bisector(model: geometor.model.Model, p1: sympy.geometry.Point, p2: sympy.geometry.Point, *, ID: str = '', classes: list[str] | None = None, guide: bool = False) → sympy.geometry.Line

Construct the perpendicular bisector of p1 p2 by Euclid I.10 symmetric circles.

Parameters:

• model – Target Model.

• p1 – First endpoint.

• p2 – Second endpoint.

• ID – Optional explicit line identifier.

• classes – Optional semantic CSS classes.

• guide – Whether the resulting bisector should remain a guide.

Returns:

Exact perpendicular bisector sympy.geometry.Line.

geometor.casey.compass_perpendicular_line(model: geometor.model.Model, line_p1: sympy.geometry.Point, line_p2: sympy.geometry.Point, through: sympy.geometry.Point, *, off_line_ref: sympy.geometry.Point | None = None, ID: str = '', classes: list[str] | None = None, guide: bool = False) → sympy.geometry.Line

Construct the perpendicular to line line_p1 line_p2 through through.

Uses rational circle reflection (Euclid I.12) when through lies off the line, and Thales’s semicircle construction (Euclid III.31) via off_line_ref (or equilateral fallback) when through lies on the line—preserving rational fields without extraneous nested radicals whenever off_line_ref is provided.

Parameters:

• model – Target Model.

• line_p1 – First point defining the base line.

• line_p2 – Second point defining the base line.

• through – Point through which the perpendicular is drawn.

• off_line_ref – Optional known point not lying on line_p1 line_p2 for Thales III.31.

• ID – Optional explicit line identifier.

• classes – Optional semantic CSS classes.

• guide – Whether the perpendicular line should remain a guide.

Returns:

Exact perpendicular sympy.geometry.Line passing through through.

geometor.casey.invert_point(pt: sympy.geometry.Point, center: sympy.geometry.Point, radius_sq: sympy.Expr) → sympy.geometry.Point

Compute the exact circular inverse of pt with respect to (center, sqrt(radius_sq)).

Implements Casey, Book III, Def. (p. 41) & Book VI, Sec. IV, Def. (p. 95): collinear with center and pt on the same ray such that |O P| * |O P'| = k^2.

Parameters:

• pt – Point P distinct from center.

• center – Centre of inversion O.

• radius_sq – Square k^2 of the radius of inversion.

Returns:

Exact inverted sympy.geometry.Point P'.

Raises:

ValueError – If pt coincides with center.

geometor.casey.polar_line(pt: sympy.geometry.Point, circ: sympy.geometry.Circle) → sympy.geometry.Line

Compute the exact polar line of pt with respect to circ (Casey Book III, Def. p. 44).

By Casey’s definition, the polar of P with respect to (O, r) is the perpendicular to ray OP erected at the inverse point P' of P. Note that sympy.geometry.Circle natively lacks polar_line and pole_of_line methods.

Parameters:

• pt – Pole P distinct from circ.center.

• circ – Reference circle (O, r).

Returns:

Exact sympy.geometry.Line representing polar(P).

Raises:

ValueError – If pt coincides with circ.center.

geometor.casey.pole_of_line(line: sympy.geometry.Line, circ: sympy.geometry.Circle) → sympy.geometry.Point

Compute the exact pole of line with respect to circ (Casey Book III, Def. p. 44).

Constructed algebraically as the circular inverse of the orthogonal foot of circ.center upon line.

Parameters:

• line – Non-diametral sympy.geometry.Line.

• circ – Reference circle (O, r).

Returns:

Exact sympy.geometry.Point whose polar with respect to circ is line.

Raises:

ValueError – If line passes through circ.center (pole is at infinity).

geometor.casey.radical_axis_line(c1: sympy.geometry.Circle, c2: sympy.geometry.Circle) → sympy.geometry.Line

Compute the exact radical axis of two non-concentric circles (Casey Book III, Prop. 23).

Parameters:

• c1 – First circle (O1, r1).

• c2 – Second circle (O2, r2).

Returns:

Exact sympy.geometry.Line locus of equal circle powers.

Raises:

ValueError – If c1 and c2 are concentric.

geometor.casey.signed_triangle_area(a_pt: sympy.geometry.Point, b_pt: sympy.geometry.Point, c_pt: sympy.geometry.Point) → sympy.Expr

Compute the exact signed area of oriented triangle ABC.

Parameters:

• a_pt – First vertex A.

• b_pt – Second vertex B.

• c_pt – Third vertex C.

Returns:

Exact symbolic signed area ((B.x - A.x)*(C.y - A.y) - (B.y - A.y)*(C.x - A.x)) / 2.

class geometor.casey.CompleteQuadrilateralResult

Constructed elements for Casey’s Complete Quadrilateral Harmonic Section (Unit 3).

model: geometor.model.Model

quad_vertices: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

opposite_meets: tuple[sympy.geometry.Point, sympy.geometry.Point]

diagonal_center: sympy.geometry.Point

third_diagonal_meets: tuple[sympy.geometry.Point, sympy.geometry.Point]

cross_ratio_ac: sympy.Expr

cross_ratio_bd: sympy.Expr

class geometor.casey.LaHireSymbolicProof

Universal symbolic proof certificate for La Hire’s Reciprocity & Harmonic Secants.

reciprocity_bilinear_diff: sympy.Expr

secant_cross_ratio: sympy.Expr

verified: bool

class geometor.casey.PolePolarResult

Constructed elements for Pole, Polar, Harmonic Secants & La Hire Reciprocity (Unit 4).

model: geometor.model.Model

circle: sympy.geometry.Circle

center: sympy.geometry.Point

pole_d: sympy.geometry.Point

secant_1_pts: tuple[sympy.geometry.Point, sympy.geometry.Point]

secant_2_pts: tuple[sympy.geometry.Point, sympy.geometry.Point]

self_conjugate_triangle: tuple[sympy.geometry.Point, sympy.geometry.Point, sympy.geometry.Point]

polar_d: sympy.geometry.Line

harmonic_meets: tuple[sympy.geometry.Point, sympy.geometry.Point]

reciprocal_pole_q: sympy.geometry.Point

polar_q: sympy.geometry.Line

cross_ratio_secant_1: sympy.Expr

cross_ratio_secant_2: sympy.Expr

lahire_verified: bool

geometor.casey.construct_complete_quadrilateral_harmonic(model: geometor.model.Model | None = None) → CompleteQuadrilateralResult

Construct Casey’s Complete Quadrilateral and verify harmonic diagonal division (Unit 3).

Follows Casey, Book VI, Sec. III, Prop. 7 (p. 91) using pure straightedge incidences on exact integer vertices A=(0, 0), B=(6, 0), C=(4, 2), D=(2, 3):

• Opposite side meets: E = [AB] ∩ [CD] = (8, 0) and F = [AD] ∩ [BC] = (12/5, 18/5).

• Internal diagonal meet: P = [AC] ∩ [BD] = (18/5, 9/5).

• Third diagonal [EF] cuts [AC] at Qa = (9/2, 9/4) and [BD] at Qb = (-6, 9) such that (A, C; P, Qa) = -1 and (B, D; P, Qb) = -1.

Parameters:

model – Optional existing Model.

Returns:

Populated CompleteQuadrilateralResult.

geometor.casey.construct_pole_and_polar(model: geometor.model.Model | None = None) → PolePolarResult

Construct the Polar of D by straightedge and verify La Hire’s Reciprocity (Unit 4).

On circle (O, R0) with O = (0, 0) and R0 = (5, 0) (x^2 + y^2 = 25):

1. Draws two rational secants cutting the circle in G1=(0, 5), H1=(4, 3) and G2=(-3, -4), H2=(5, 0), meeting at exterior pole D = (15/2, 5/4).

2. Constructs the remaining diagonal vertices U = [G1 H2] ∩ [G2 H1] = (3, 2) and V = [G1 G2] ∩ [H1 H2] = (5/3, 10) of the inscribed quadrangle; their join [U V] is polar(D): 6x + y - 20 = 0, and D U V is a Self-Conjugate Triangle (Casey, Book VI, Sec. VII, Prop. 2, p. 142).

3. Intersects polar(D) with both secants at J1, J2 and verifies Casey’s Harmonic Secant Theorem (G1, H1; J1, D) = (G2, H2; J2, D) = -1 (Sec. III, Prop. 4, p. 89).

4. Selects exterior point Q = (4, -4) on polar(D), constructs its two tangents to (O, 5) by Thales semicircle on OQ, draws chord of contact polar(Q) (Book III, Prop. 27, p. 47), and asserts La Hire’s Reciprocity D in polar(Q) (Book III, Prop. 25, p. 45).

Parameters:

model – Optional existing Model.

Returns:

Populated PolePolarResult.

geometor.casey.verify_lahire_reciprocity_symbolic() → LaHireSymbolicProof

Prove La Hire’s Reciprocity (Book III, Prop. 25) and Harmonic Secant Section (VI.III.4).

1. For circle (x - h)^2 + (y - k)^2 = r^2, polar(P) has equation (x1 - h)(x - h) + (y1 - k)(y - k) - r^2 = 0, which is symmetric in (x1, y1) and (x2, y2).

2. For exterior pole D = (d, 0) on diameter secant y = 0 cutting x^2 + y^2 = r^2 at G = (-r, 0), H = (r, 0) with polar foot J = (r^2/d, 0), the anharmonic ratio (G, H; J, D) simplifies identically to -1.

Returns:

Frozen LaHireSymbolicProof certificate.

geometor.casey.EMBEDDED_SVG_STYLE: str = ''

geometor.casey.export_svg(model: geometor.model.Model, output_path: str | pathlib.Path, *, graphic_id: str = 'g1', title: str = 'Casey Euclidean Circle Construction', width: int = 680, height: int = 480, margin: float = 42.0, bbox: tuple[float, float, float, float] | None = None, focus_points: collections.abc.Sequence[sympy.geometry.Point] | None = None, include_guides: bool = False, show_labels: bool = True, label_overrides: dict[str, tuple[str, str]] | None = None, highlight_point_ids: collections.abc.Sequence[str] | None = None) → pathlib.Path

Write an interactive .euclid-graphic SVG rendering of model to output_path.

Parameters:

• model – Populated Model.

• output_path – Destination .svg file path.

• graphic_id – data-graphic-id identifier for geometor_interactive.js.

• title – Accessible SVG title string.

• width – Pixel width of the SVG viewBox.

• height – Pixel height of the SVG viewBox.

• margin – Outer pixel margin around the framed viewport.

• bbox – Optional explicit math viewport (xmin, xmax, ymin, ymax).

• focus_points – Optional subset of points used to compute auto-framing bounds.

• include_guides – If True, also render elements marked guide=True.

• show_labels – Master toggle for point text labels.

• label_overrides – Optional mapping pt_id -> (base, subscript).

• highlight_point_ids – Optional point IDs to emphasize with an outer halo ring.

Returns:

Resolved pathlib.Path of the written SVG file.

geometor.casey.render_svg_string(model: geometor.model.Model, *, graphic_id: str = 'g1', title: str = 'Casey Euclidean Circle Construction', width: int = 680, height: int = 480, margin: float = 42.0, bbox: tuple[float, float, float, float] | None = None, focus_points: collections.abc.Sequence[sympy.geometry.Point] | None = None, include_guides: bool = False, show_labels: bool = True, label_overrides: dict[str, tuple[str, str]] | None = None, highlight_point_ids: collections.abc.Sequence[str] | None = None) → str

Render a Model into a complete .euclid-graphic SVG string.

Parameters:

• model – Populated Model.

• graphic_id – data-graphic-id identifier for geometor_interactive.js.

• title – Accessible SVG title string.

• width – Pixel width of the SVG viewBox.

• height – Pixel height of the SVG viewBox.

• margin – Outer pixel margin around the framed viewport.

• bbox – Optional explicit math viewport (xmin, xmax, ymin, ymax).

• focus_points – Optional subset of points used to compute auto-framing bounds.

• include_guides – If True, also render elements marked guide=True.

• show_labels – Master toggle for point text labels.

• label_overrides – Optional mapping pt_id -> (base, subscript).

• highlight_point_ids – Optional point IDs to emphasize with an outer halo ring.

Returns:

UTF-8 XML/SVG string.

geometor.casey.__version__ = '0.1.0'