← casey / A Sequel to the First Six Books of the Elements of Euclid

Technical Glossary of Circle, Projective & Triangle Geometry¶

30 Foundational Definitions & Invariants in Casey’s Sequel to Euclid

Every :term: reference across the 3-Track Book Codex resolves to its technical definition below.

Anharmonic Ratio (Cross-Ratio)¶

Terminology / Origin: English / French (rapport anharmonique)
Algebraic / Synthetic Form: (ABCD) = (AC·BD)/(AD·BC)
Definition: Projective invariant of four collinear points or four concurrent rays introduced in this terminology by Michel Chasles and treated in Casey Book VI, Sec. VI (pp. 127–140). Equal to -1 in the case of harmonic section.

Antiparallel¶

Terminology / Origin: English / Greek
Algebraic / Synthetic Form: Transversal cutting two sides at transposed base angles
Definition: A straight line cutting AB, AC at D, E so that angle ADE = angle ACB and angle AED = angle ABC (Casey Supp. Ch., Sec. I, Def., p. 169). Parallel to the tangent to the circumcircle at A and bisected by the symmedian from A.

Brocard Angle (ω)¶

Terminology / Origin: English
Algebraic / Synthetic Form: cot ω = cot A + cot B + cot C = (a² + b² + c²) / (4Δ)
Definition: Common acute angle ω = ∠ΩAB = ∠ΩBC = ∠ΩCA = ∠Ω’BA = ∠Ω’CB = ∠Ω’AC of a triangle ABC (Casey Supp. Ch., Sec. II, pp. 176–177), bounded above by 30° with equality iff ABC is equilateral.

Brocard Circle¶

Terminology / Origin: English
Algebraic / Synthetic Form: Seven-point circle on diameter OK
Definition: The circle having the segment joining the circumcentre O = X(3) and symmedian point K = X(6) as diameter (Casey Supp. Ch., Sec. V, pp. 194–195). Passes through O, K, both Brocard points Ω and Ω’, and the three vertices A’, B’, C’ of Brocard’s First Triangle.

Brocard Points (Ω, Ω’)¶

Terminology / Origin: English
Algebraic / Synthetic Form: Isogonal bicentric pair with barycentrics (1/b² : 1/c² : 1/a²) and (1/c² : 1/a² : 1/b²)
Definition: Unique interior points Ω (first) and Ω’ (second) at which the sides of ABC subtend equal oriented angles ω = ∠ΩAB = ∠ΩBC = ∠ΩCA and ω = ∠Ω’BA = ∠Ω’CB = ∠Ω’AC (Casey Supp. Ch., Sec. II, pp. 175–178).

Casey’s Theorem (Generalized Ptolemy Theorem)¶

Terminology / Origin: English
Algebraic / Synthetic Form: t₁₂·t₃₄ + t₁₄·t₂₃ = t₁₃·t₂₄
Definition: John Casey’s 1866 theorem (Book VI, Sec. IV, Prop. 10, pp. 103–104): four circles 1, 2, 3, 4 are tangent to a fifth circle (or straight line) iff their pairwise common tangents satisfy 12·34 ± 14·23 ± 13·24 = 0. Reduces to Ptolemy’s Theorem (Euclid VI.D) when all four radii vanish.

Centre of Mean Position (Mean Centre)¶

Terminology / Origin: English
Algebraic / Synthetic Form: Weighted barycentre Σ mᵢ Pᵢ / Σ mᵢ
Definition: Unique point G associated to weighted points Pᵢ with weights mᵢ such that for every line L, Σ mᵢ·dist(Pᵢ, L) = (Σ mᵢ)·dist(G, L) (Casey Book I, Props. 17–18, pp. 12–13).

Centre of Similitude (Homothetic Centre)¶

Terminology / Origin: English
Algebraic / Synthetic Form: External and internal homothetic centres dividing OO' in ratio r₁ : ±r₂
Definition: Points on the line of centres of two circles through which all direct (external) or transverse (internal) parallel-radius joins and common tangents pass (Casey Book VI, Sec. II, pp. 82–87).

Coaxal System of Circles¶

Terminology / Origin: English
Algebraic / Synthetic Form: One-parameter pencil of circles sharing a common radical axis
Definition: Family of circles every pair of which has the same radical axis (Casey Book III, Prop. 24, p. 44; Book VI, Sec. V, pp. 113–119). Classified into intersecting (common chord), non-intersecting (real Poncelet limiting points L₁, L₂), and tangent pencils.

Complete Quadrilateral¶

Terminology / Origin: English
Algebraic / Synthetic Form: Four general lines meeting in six vertices with three diagonals
Definition: Projective figure formed by four coplanar straight lines in general position, possessing six vertices in three opposite pairs and three diagonals each divided harmonically by the other two (Casey Book VI, Sec. III, Prop. 7, p. 91).

Cosine Circle (Second Lemoine Circle)¶

Terminology / Origin: English
Algebraic / Synthetic Form: Six-point circle centered at K cut by antiparallels through K
Definition: Circle passing through the six intersections of the antiparallels through the symmedian point K with the sides of ABC (Casey Supp. Ch., Sec. III, Prop. 1 Cor. 4, p. 183). Centered at K with side intercepts proportional to cos A : cos B : cos C.

Euler Line¶

Terminology / Origin: English
Algebraic / Synthetic Form: Collinear axis containing O = X(3), G = X(2), N = X(5), H = X(4), and L = X(20)
Definition: Straight line on which lie the circumcentre O, centroid G, nine-points centre N, orthocentre H, and de Longchamps point L, with HG = 2·GO and HN = NO (Casey Book I, Misc. Ex. 43, p. 20; Book IV, Prop. 5).

Feuerbach Point (X(11))¶

Terminology / Origin: English
Algebraic / Synthetic Form: Internal contact point of the Nine-Points Circle and Incircle
Definition: Point Fe = X(11) at which the Nine-Points Circle touches the Incircle internally (Casey Book VI, Sec. IV, Prop. 10 Cor. 2, pp. 104–106). On the canonical Heronian (15, 13, 14) triangle, Fe = (6, 8).

Harmonic Polygon¶

Terminology / Origin: English
Algebraic / Synthetic Form: Cyclic n-gon which is the projection/inversion of a regular n-gon
Definition: Cyclic polygon inscribed in a circle possessing a generalized symmedian/Lemoine point whose distances to the sides are proportional to the side lengths (Casey 1886; Supp. Ch., Sec. VI, pp. 205–218).

Harmonic Section / Harmonic Range¶

Terminology / Origin: English
Algebraic / Synthetic Form: Internal and external division in the same ratio: (A, B; C, D) = -1
Definition: Division of segment AB internally at C and externally at D so that AC : CB = AD : DB, equivalently AB is the harmonic mean of AC and AD (Casey Book VI, Sec. III, Def., p. 87).

Hart’s Circle¶

Terminology / Origin: English
Algebraic / Synthetic Form: Circular-triangle generalization of the Nine-Points Circle
Definition: Andrew Searle Hart’s 1861 theorem (Casey Book VI, Sec. IV, Prop. 11, pp. 106–108): for any circular triangle formed by three circles, the eight circles touching all three partition into sets of four that are each touched by a fourth circle (Hart’s Circle).

Inversion (Circular Inversion)¶

Terminology / Origin: English
Algebraic / Synthetic Form: Conformal involutory map P -> P' on ray OP with OP·OP' = k²
Definition: Transformation with respect to a circle of inversion (O, k) sending each point P != O to P’ on ray OP with OP·OP’ = k² (Casey Book III, Def. p. 41; Book VI, Sec. IV, pp. 95–112). Preserves angles and cross-ratios while interchanging lines through O and lines/circles.

Isogonal Conjugates¶

Terminology / Origin: English
Algebraic / Synthetic Form: Cevian pair symmetric across the internal angle bisectors
Definition: Two points P, P’ whose rays from A, B, C are reflections of one another across the angle bisectors of ABC (Casey Supp. Ch., Sec. I, Prop. 1, p. 165). Their six pedal projections onto the sides lie on a single pedal circle centered at the midpoint of PP’.

Isotomic Conjugates¶

Terminology / Origin: English
Algebraic / Synthetic Form: Cevian pair cutting each side at points equidistant from the side midpoint
Definition: Points whose cevians meet BC, CA, AB in points symmetric with respect to the side midpoints Ma, Mb, Mc (Casey Supp. Ch., Sec. I, Def., p. 167).

Lemoine’s First Circle (Triplicate-Ratio Circle)¶

Terminology / Origin: English
Algebraic / Synthetic Form: Six-point circle cut on ABC by parallels to the sides through K
Definition: Circle passing through the six intersections of the three parallels through the symmedian point K to the sides of ABC (Casey Supp. Ch., Sec. III, Prop. 1, pp. 179–181). Centered at the midpoint of OK = X(182), with side intercepts proportional to a³ : b³ : c³.

Limiting Points (Poncelet Points)¶

Terminology / Origin: English / French (points limites)
Algebraic / Synthetic Form: Zero-radius circles L₁, L₂ of a non-intersecting coaxal pencil
Definition: Two points L₁, L₂ on the line of centres on opposite sides of the radical axis at distance equal to the tangent from the radical foot D to any circle of the system (Casey Book III, Prop. 23 Def., p. 43). Every circle of the conjugate orthogonal pencil passes through L₁ and L₂, and L₁, L₂ are inverse points with respect to every circle of the coaxal pencil.

Nine-Points Circle¶

Terminology / Origin: English
Algebraic / Synthetic Form: Circle (N, R/2) through midpoints, altitude feet, and Euler midpoints
Definition: Circle centered at N = X(5) = midpoint(O, H) of radius R/2 passing through the three side midpoints, the three altitude feet, and the three midpoints of segments joining H to the vertices (Casey Book IV, Prop. 5, pp. 57–58).

Orthogonal Circles¶

Terminology / Origin: English
Algebraic / Synthetic Form: Circles intersecting at right angles: |O₁O₂|² = r₁² + r₂²
Definition: Two circles whose tangents (and hence radii) at each point of intersection are perpendicular, characterized algebraically by d² = r₁² + r₂² (Casey Book III, Prop. 22, p. 42). Any circle orthogonal to (O, k) is mapped onto itself by inversion in (O, k).

Pole and Polar¶

Terminology / Origin: English
Algebraic / Synthetic Form: Projective point-line duality induced by a circle: (x₀-h)(x-h)+(y₀-k)(y-k)=r²
Definition: For a point P != O and circle (O, r), the polar of P is the perpendicular to OP at the inverse point P’ of P (Casey Book III, Def., p. 44). Governed by La Hire’s Reciprocity (Prop. 25): Q lies on polar(P) iff P lies on polar(Q).

Power of a Point¶

Terminology / Origin: English / German (Potenz eines Punktes)
Algebraic / Synthetic Form: |PO|² - r² = PA · PB
Definition: Jakob Steiner’s invariant PA·PB = PO² - r² for any secant PAB through P meeting circle (O, r) at A, B (Euclid III.35–36; Casey Book III, Props. 20–24, pp. 41–44).

Radical Axis & Radical Centre¶

Terminology / Origin: English / French (axe radical)
Algebraic / Synthetic Form: Linear locus of equal circle powers pow(P, C₁) = pow(P, C₂)
Definition: Straight line perpendicular to the line of centres along which tangents (or powers) with respect to two non-concentric circles are equal (Casey Book III, Prop. 23, p. 43). For three circles with non-collinear centres, the three pairwise radical axes concur at the Radical Centre.

Reciprocation (Polar Duality)¶

Terminology / Origin: English
Algebraic / Synthetic Form: Duality mapping points to polars and lines to poles
Definition: Method of deriving dual projective theorems by replacing every point by its polar and every line by its pole with respect to an auxiliary base circle (Casey Book VI, Sec. VII, pp. 141–147), transforming Pascal’s inscribed hexagon into Brianchon’s circumscribed hexagon.

Steiner Point (X(99)) & Tarry Point (X(98))¶

Terminology / Origin: English
Algebraic / Synthetic Form: Antipodal circumcircle pair associated with Brocard's First Triangle
Definition: Parallels through A, B, C to the corresponding sides of Brocard’s First Triangle concur at Steiner’s Point X(99) on the circumcircle; perpendiculars concur at its diametral antipode, Tarry’s Point X(98) (Casey Supp. Ch., Sec. VII).

Symmedian & Lemoine’s Point (K = X(6))¶

Terminology / Origin: English / French (symédiane)
Algebraic / Synthetic Form: Isogonal reflection of a median; concurrency K = (a² : b² : c²)
Definition: Isogonal conjugate of a median across the internal angle bisector. The three symmedians concur at Lemoine’s Point K = X(6), whose perpendicular distances to the sides are proportional to the side lengths and minimize x² + y² + z² (Casey Supp. Ch., Sec. I, Props. 5–7, pp. 170–172).

Tucker’s Circles & Taylor’s Circle¶

Terminology / Origin: English
Algebraic / Synthetic Form: One-parameter family of six-point antiparallel/parallel triangle circles
Definition: Family of circles cut on the sides of ABC by triangles homothetic to ABC with homothetic centre K, centered along OK (Casey Supp. Ch., Sec. III, pp. 181–186). Includes the Circumcircle, First Lemoine Circle, Cosine Circle, and Taylor’s Circle (six projections of altitude feet onto the adjacent sides).