Technical Glossary of Modern Synthetic Geometry

Analysis (Step I of Construction)¶

Domain & Locus: Constructive Method (§ 4)
Symbolic / Center Identity: Step 1 of 4-Part Pattern
Edition Loci: f. 0022 (p. 3)

First stage of Altshiller-Court’s canonical four-part solution of a construction problem: assume the problem solved, draw a tentative sketch containing both given and required elements, and deduce auxiliary relations that uniquely locate the unknown elements.

analytic method (geometric analysis)¶

Domain & Locus: Constructive Method (§ 4)
Symbolic / Center Identity: Method of Resolution (ἀνάλυσις)
Edition Loci: f. 0022 (p. 3)

Regressive method of solving a construction problem or discovering a proof by assuming the required figure already constructed, studying the geometric relations among its given and sought elements, and reducing the problem step-by-step to previously known elementary constructions.

anticenter of a cyclic quadrilateral¶

Domain & Locus: Quadrilateral Geometry (§ 260)
Symbolic / Center Identity: Point symmetric to circumcenter across centroid
Edition Loci: f. 0151 (p. 132)

Point of concurrency of the four perpendiculars dropped from the midpoint of each side of a cyclic quadrilateral to the opposite side (called ‘mathothetic’ / Mathot point).

anticomplementary triangle¶

Domain & Locus: Triangle Geometry (§ 99)
Symbolic / Center Identity: Triangle through A, B, C parallel to opposite sides
Edition Loci: f. 0087 (p. 68)

Triangle formed by the lines through vertices A, B, C parallel to BC, CA, AB respectively; homothetic to ABC from the centroid G = X(2) with ratio -2.

Apollonian circle¶

Domain & Locus: Locus & Triangle Geometry (§ 11, § 61, § 597)
Symbolic / Center Identity: Circle of Apollonius
Edition Loci: f. 0034 (p. 15) · f. 0035 (p. 16) · f. 0036 (p. 17) · f. 0039 (p. 20)

The locus of a point whose distances from two fixed points A, B have a constant ratio m : n (m ≠ n); its diameter joins the two points C, D that divide AB internally and externally in the ratio m : n (§ 11, Locus 11; § 61). For a triangle ABC, the three Apollonian circles pass through each vertex and the feet of the internal and external angle bisectors on the opposite side (§ 597), intersecting in the two isodynamic points X(15), X(16).

Brocard circle¶

Domain & Locus: Brocard Geometry (§ 666)
Symbolic / Center Identity: Seven-point circle on diameter OK, center X(182)
Edition Loci: f. 0298 (p. 279)

Circle having the segment joining the circumcenter O = X(3) and Lemoine point K = X(6) as diameter; passes through O, K, the two Brocard points Ω, Ω’, and the three vertices of Brocard’s first triangle.

Brocard points and Brocard angle¶

Domain & Locus: Brocard Geometry (§ 650, § 654)
Symbolic / Center Identity: Ω, Ω' and angle ω: cot ω = cot A + cot B + cot C
Edition Loci: f. 0293 (p. 274)

Unique interior points Ω (direct) and Ω’ (retrograde), isogonal conjugates of each other, satisfying ∠ΩBC = ∠ΩCA = ∠ΩAB = ω and ∠Ω’CB = ∠Ω’BA = ∠Ω’AC = ω, where cot ω = cot A + cot B + cot C = (a² + b² + c²) / (4Δ).

capable arc (locus of constant subtended angle)¶

Domain & Locus: Locus Geometry (§ 11, Locus 7)
Symbolic / Center Identity: Arc AMB subtending given ∠α on chord AB
Edition Loci: f. 0031 (p. 12)

Set of points on a given side of segment AB at which AB subtends a given angle α (Euclid III.21, III.33); when α = 90°, the two symmetric arcs unite into the circle on diameter AB (Thales’ locus, Euclid III.31).

centers of similitude¶

Domain & Locus: Circle & Homothecy Geometry (§ 39, § 394)
Symbolic / Center Identity: Direct (external) and inverse (internal) homothetic centers
Edition Loci: f. 0060 (p. 41)

Two circles of unequal radii are homothetic in two ways: via their external (direct) center of similitude (positive ratio +R/R’) and their internal (inverse) center of similitude (negative ratio -R/R’), which divide the line of centers harmonically. For the circumcircle and incircle of a triangle, these correspond to Kimberling centers X(56) and X(55).

centroid (center of gravity)¶

Domain & Locus: Triangle Geometry (§ 1, § 89)
Symbolic / Center Identity: G = X(2)
Edition Loci: f. 0085 (p. 66)

Point of concurrency of the three medians AA’, BB’, CC’ of a triangle, trisecting each median in the ratio AG : GA’ = 2 : 1 (Kimberling center X(2)).

cevian¶

Domain & Locus: Triangle Geometry (§ 326)
Symbolic / Center Identity: Line from a vertex to the opposite sideline
Edition Loci: f. 0178 (p. 159)

Any line segment or line joining a vertex of a triangle to a point on the opposite sideline (medians, altitudes, angle bisectors, and symmedians are special cevians).

circular inversion¶

Domain & Locus: Transformation Geometry (§ 518)
Symbolic / Center Identity: (O, k): OP · OP' = k
Edition Loci: f. 0249 (p. 230)

Involutory conformal point transformation mapping P ≠ O to P’ on ray OP (or opposite ray when k < 0) with OP · OP’ = k; maps lines and circles not through O to circles, and circles through O to straight lines.

circumcircle and circumcenter¶

Domain & Locus: Triangle Geometry (§ 1, § 68)
Symbolic / Center Identity: (O), radius R, center O = X(3)
Edition Loci: f. 0020 (p. 1)

Unique circle passing through the three vertices A, B, C of a triangle; its center O (Kimberling X(3)) is the intersection of the three perpendicular bisectors of the sides.

coaxal (coaxial) pencil of circles¶

Domain & Locus: Circle Geometry (§ 440)
Symbolic / Center Identity: Intersecting, tangent, and non-intersecting pencils; limiting points
Edition Loci: f. 0220 (p. 201)

One-parameter family of circles every pair of which shares the same radical axis; a non-intersecting pencil possesses two point-circles (Poncelet’s limiting points L, L’) that are inverse with respect to every circle of the pencil, while the orthogonal circles form the conjugate coaxal pencil through L and L’.

complementary (medial) triangle¶

Domain & Locus: Triangle Geometry (§ 96)
Symbolic / Center Identity: Medial triangle A'B'C'
Edition Loci: f. 0087 (p. 68)

Triangle formed by the midpoints A’, B’, C’ of the sides BC, CA, AB of reference triangle ABC; homothetic to ABC from the centroid G = X(2) with ratio -1/2.

complete quadrangle and complete quadrilateral¶

Domain & Locus: Projective Geometry (§ 354)
Symbolic / Center Identity: Dual four-point and four-line configurations
Edition Loci: f. 0189 (p. 170)

A complete quadrangle consists of four points (no three collinear) and their six joining sides meeting in three diagonal points; dual to a complete quadrilateral of four lines and six vertices, generating harmonic ranges automatically via straightedge alone.

Construction (Step II of Construction)¶

Domain & Locus: Constructive Method (§ 4)
Symbolic / Center Identity: Step 2 of 4-Part Pattern
Edition Loci: f. 0022 (p. 3)

Second stage of the four-part constructive pattern: state the explicit ruler-and-compass steps derived from the Analysis that actually produce the required geometric figure from the given data.

directed (signed) segments¶

Domain & Locus: Projective & Metric Geometry (§ 306)
Symbolic / Center Identity: AB = -BA; Chasles' relation AB + BC + CA = 0
Edition Loci: f. 0170 (p. 151)

Segments on an oriented line endowed with algebraic sign according to sense, unifying internal and external division formulas across Menelaus’, Ceva’s, and harmonic theorems.

Discussion (Step IV of Construction)¶

Domain & Locus: Constructive Method (§ 4)
Symbolic / Center Identity: Step 4 of 4-Part Pattern (διορισμός)
Edition Loci: f. 0022 (p. 3)

Fourth stage of the four-part constructive pattern (Greek diorismos): determine the exact conditions on the given data for a solution to exist, count the number of distinct solutions, and examine limiting or degenerate configurations.

Euler line¶

Domain & Locus: Triangle Geometry (§ 202)
Symbolic / Center Identity: Collinearity of O, G, N, H
Edition Loci: f. 0121 (p. 102)

Line containing the circumcenter O = X(3), centroid G = X(2), nine-point center F/N = X(5), and orthocenter H = X(4) of any non-equilateral triangle, satisfying OG : GH = 1 : 2 and ON : NH = 1 : 1.

Feuerbach’s theorem and Feuerbach point¶

Domain & Locus: Triangle Geometry (§ 215, § 540)
Symbolic / Center Identity: Tangency of Nine-Point Circle with Tritangent Circles; X(11)
Edition Loci: f. 0124 (p. 105)

Theorem (Feuerbach, 1822) that the nine-point circle of a triangle is internally tangent to the incircle (at the Feuerbach point X(11)) and externally tangent to each of the three excircles.

fourth proportional and mean proportional¶

Domain & Locus: Basic Constructions (§ 2)
Symbolic / Center Identity: a : b = c : x and a : x = x : b
Edition Loci: f. 0020 (p. 1) · f. 0021 (p. 2)

Classical Euclidean ruler-and-compass proportion constructions (Elements VI.12 and VI.13): given three segments a, b, c, the fourth proportional satisfies x = bc/a via parallel intercepts; given two segments p, q, the mean (geometric) proportional satisfies x² = pq via the altitude to the hypotenuse of a semicircle.

geometric locus¶

Domain & Locus: Constructive Method (§ 11)
Symbolic / Center Identity: Method of Loci
Edition Loci: f. 0030 (p. 11)

Set of all points (and only those points) satisfying a specified geometric condition. In the Method of Loci, an unknown point is constructed as the intersection of two independent loci obtained by setting aside one condition at a time.

Gergonne point¶

Domain & Locus: Triangle Centers (§ 331)
Symbolic / Center Identity: X(7)
Edition Loci: f. 0179 (p. 160)

Point of concurrency of the three cevians joining the vertices of a triangle to the points of contact of the inscribed circle with the opposite sides (Kimberling center X(7)).

harmonic division and harmonic conjugates¶

Domain & Locus: Projective Geometry (§ 59, § 345)
Symbolic / Center Identity: (ABCD) = -1
Edition Loci: f. 0006 (p. vii) · f. 0034 (p. 15) · f. 0185 (p. 166)

Four collinear points A, B, C, D form a harmonic range when C and D divide AB internally and externally in the same numerical ratio, so (AC/CB) : (AD/DB) = -1.

homothecy (homothety)¶

Domain & Locus: Transformation Geometry (§ 31)
Symbolic / Center Identity: Central dilation / Homothetic transformation
Edition Loci: f. 0057 (p. 38)

Transformation (also spelled homothety) mapping each point M to M’ collinear with a fixed homothetic center S (center of similitude) such that SM’ / SM = k is a non-zero signed constant (the homothetic ratio).

incircle, inradius, and incenter¶

Domain & Locus: Triangle Geometry (§ 1, § 116)
Symbolic / Center Identity: (I, r), radius r, center I = X(1)
Edition Loci: f. 0020 (p. 1)

Circle internally tangent to all three sides of a triangle ABC; its center I (the incenter, Kimberling center X(1)) is the point of concurrency of the three internal angle bisectors, equidistant by the inradius r from the three sides.

indirect elements¶

Domain & Locus: Constructive Method (§§ 15–22)
Symbolic / Center Identity: Sums, differences, and auxiliary elements
Edition Loci: f. 0040 (p. 21)

Composite or derived data given in place of primary triangle elements—such as the perimeter 2s, sum or difference of two sides (b + c, b - c), sum or difference of angles, medians, altitudes, or radii—translated into constructible auxiliary triangles.

inverse points with respect to a circle¶

Domain & Locus: Circle Geometry (§ 358)
Symbolic / Center Identity: OP · OP' = R²
Edition Loci: f. 0191 (p. 172)

Two points P, P’ on the same ray from the center O of a circle of radius R satisfying OP · OP’ = R²; every circle through P and P’ cuts the reference circle orthogonally (§ 366).

isodynamic points¶

Domain & Locus: Triangle Centers (§ 606)
Symbolic / Center Identity: W, W' = X(15), X(16)
Edition Loci: f. 0034 (p. 15) · f. 0039 (p. 20) · f. 0281 (p. 262)

The two common intersections of the three Apollonian circles of a triangle (Kimberling centers X(15) and X(16)); inverse to one another in the circumcircle, collinear with O and K on the Brocard axis, and having equilateral pedal triangles.

isogonal lines and isogonal conjugates¶

Domain & Locus: Modern Triangle Geometry (§ 625, § 632)
Symbolic / Center Identity: Reflection across angle bisectors
Edition Loci: f. 0286 (p. 267)

Two rays through a vertex symmetric about the angle bisector are isogonal; if cevians to P concur, their isogonals concur at the isogonal conjugate P’ (e.g., orthocenter H = X(4) ↔ circumcenter O = X(3); centroid G = X(2) ↔ Lemoine point K = X(6); incenter I = X(1) self-conjugate).

isotomic points and isotomic conjugate¶

Domain & Locus: Triangle Geometry (§ 329, § 334)
Symbolic / Center Identity: Symmetric division about side midpoints
Edition Loci: f. 0179 (p. 160)

Two points on a triangle side equidistant from its midpoint are isotomic; if cevians to three points concur at P, the cevians to their isotomic points concur at the isotomic conjugate P’ (e.g., Gergonne X(7) and Nagel X(8)).

Lemoine circles (first and second/cosine)¶

Domain & Locus: Lemoine Geometry (§§ 588–596)
Symbolic / Center Identity: Concyclic six-point intersections through K = X(6)
Edition Loci: f. 0276 (p. 257)

The First Lemoine Circle cuts the sides in six points formed by lines through K parallel to the sides; the Second Lemoine (Cosine) Circle cuts the sides in six points formed by antiparallels through K, with chord lengths proportional to cos A, cos B, cos C.

Lemoine point (symmedian point)¶

Domain & Locus: Triangle Centers (§ 572)
Symbolic / Center Identity: K = X(6)
Edition Loci: f. 0271 (p. 252)

Point of concurrency of the three symmedians of a triangle (Kimberling center X(6)); minimizes the sum of squares of perpendicular distances to the three sides (§ 581), with distances proportional to the side lengths a, b, c.

mediator (perpendicular bisector)¶

Domain & Locus: Triangle & Locus Geometry (§ 11)
Symbolic / Center Identity: Perpendicular bisector of a segment
Edition Loci: f. 0031 (p. 12) · f. 0037 (p. 18) · f. 0040 (p. 21)

The line perpendicular to a given segment AB at its midpoint—the locus of all points equidistant from A and B (§ 11, Locus 3; Euclid I.10, I.11). The three mediators of a triangle concur at the circumcenter O = X(3).

method of similitude¶

Domain & Locus: Transformation Geometry (§ 23)
Symbolic / Center Identity: Scaling / Similarity method
Edition Loci: f. 0053 (p. 34)

Constructive technique that temporarily disregards one linear metric condition to construct any figure of the required shape (species), and then scales it to the required size by homothecy or fourth proportionals.

Nagel point¶

Domain & Locus: Triangle Centers (§ 333)
Symbolic / Center Identity: X(8)
Edition Loci: f. 0179 (p. 160)

Point of concurrency of the three cevians joining the vertices of a triangle to the points of contact of the corresponding excircles with the opposite sides (Kimberling center X(8)); isotomic conjugate of the Gergonne point X(7).

nine-point circle¶

Domain & Locus: Triangle Geometry (§ 206)
Symbolic / Center Identity: Euler / Feuerbach / Terquem circle, center X(5)
Edition Loci: f. 0122 (p. 103)

Circle of radius R/2 centered at the midpoint X(5) of OH that passes through nine classic points: the three side midpoints, the three altitude feet, and the three Euler points (midpoints of AH, BH, CH).

orthic triangle¶

Domain & Locus: Triangle Geometry (§ 183)
Symbolic / Center Identity: Pedal triangle DEF of the orthocenter
Edition Loci: f. 0116 (p. 97)

Triangle DEF whose vertices are the feet of the altitudes of ABC; for an acute triangle, its incenter is H and it solves Fagnano’s minimum-perimeter inscribed triangle problem (§ 194).

orthocenter¶

Domain & Locus: Triangle Geometry (§ 1, § 176)
Symbolic / Center Identity: H = X(4)
Edition Loci: f. 0113 (p. 94)

Point of concurrency of the three altitudes AD, BE, CF of a triangle ABC (Kimberling center X(4)); any of the four points A, B, C, H is the orthocenter of the triangle formed by the other three.

orthocentric group (orthocentric quadrilateral)¶

Domain & Locus: Triangle Geometry (§ 220)
Symbolic / Center Identity: Four points each the orthocenter of the other three
Edition Loci: f. 0128 (p. 109)

Quadruple of points A, B, C, H in which each point is the orthocenter of the triangle formed by the remaining three; all four component triangles share the exact same nine-point circle (§ 224).

orthogonal circles¶

Domain & Locus: Circle Geometry (§ 363)
Symbolic / Center Identity: r₁² + r₂² = d²
Edition Loci: f. 0193 (p. 174)

Two intersecting circles whose tangents (and radii) at each point of intersection are perpendicular.

orthopole of a line¶

Domain & Locus: Modern Triangle Geometry (§ 690)
Symbolic / Center Identity: Neuberg's concurrency of perpendiculars from projections
Edition Loci: f. 0306 (p. 287)

Given a line m and triangle ABC, if A’, B’, C’ are the orthogonal projections of A, B, C onto m, then the perpendiculars from A’, B’, C’ to BC, CA, AB respectively concur at a single point M: the orthopole of line m for triangle ABC.

pole and polar¶

Domain & Locus: Circle & Projective Geometry (§ 374)
Symbolic / Center Identity: Reciprocal point-line duality w.r.t. a circle
Edition Loci: f. 0197 (p. 178)

For a point P and circle (O, R), the polar of P is the line perpendicular to OP through the inverse point P’ of P; enjoys La Hire’s reciprocity: Q lies on the polar of P iff P lies on the polar of Q (§ 379).

power of a point with respect to a circle¶

Domain & Locus: Circle Geometry (§ 412)
Symbolic / Center Identity: d² - R² = PA · PB
Edition Loci: f. 0210 (p. 191)

Constant signed product PA · PB of the directed segments from P to the intersections A, B of any secant through P with circle (O, R), equal to PO² - R² (Steiner, 1826).

Problem of Apollonius¶

Domain & Locus: Circle Constructions (§ 506)
Symbolic / Center Identity: Circle tangent to three given circles (PPP..CCC)
Edition Loci: f. 0241 (p. 222)

Celebrated problem of constructing a circle tangent to three given elements (points, lines, or circles—ten cases I–X), culminating in Gergonne’s simultaneous construction of complementary pairs of up to eight tangent circles (§ 517).

Proof (Step III of Construction)¶

Domain & Locus: Constructive Method (§ 4)
Symbolic / Center Identity: Step 3 of 4-Part Pattern
Edition Loci: f. 0022 (p. 3)

Third stage of the four-part constructive pattern: demonstrate deductively that the figure produced by the Construction steps genuinely satisfies every condition demanded in the problem statement.

radical axis and radical center¶

Domain & Locus: Circle Geometry (§ 422, § 492)
Symbolic / Center Identity: Locus of equal powers
Edition Loci: f. 0029 (p. 10) · f. 0035 (p. 16) · f. 0213 (p. 194)

The radical axis of two non-concentric circles is the straight line of points having equal power with respect to both circles; the three radical axes of three circles taken in pairs concur at their radical center (§ 492).

Simson line (Wallace-Simson line)¶

Domain & Locus: Triangle & Circle Geometry (§ 283)
Symbolic / Center Identity: Collinear pedal projections of a circumcircle point
Edition Loci: f. 0160 (p. 141)

Line containing the feet of the perpendiculars dropped from any point P on the circumcircle of triangle ABC onto its three sidelines BC, CA, AB; bisects the segment PH joining P to the orthocenter (§ 299).

Steiner point and Tarry point¶

Domain & Locus: Triangle Centers (§ 679, § 680)
Symbolic / Center Identity: Steiner R = X(99), Tarry N = X(98)
Edition Loci: f. 0301 (p. 282)

Diametrically opposite points on the circumcircle associated with Brocard’s first triangle: parallels through A, B, C to the sides of Brocard’s first triangle concur at the Steiner point X(99), while perpendiculars concur at the Tarry point X(98).

symmedian¶

Domain & Locus: Lemoine Geometry (§ 558)
Symbolic / Center Identity: Reflection of median across angle bisector
Edition Loci: f. 0266 (p. 247)

Line through a vertex of a triangle symmetric to the median with respect to the internal bisector of that vertex angle; locus of midpoints of antiparallels to the opposite side.

synthetic method¶

Domain & Locus: Constructive Method (§ 4)
Symbolic / Center Identity: Method of Composition (σύνθεσις)
Edition Loci: f. 0022 (p. 3)

Progressive exposition that starts from elementary given data and executes constructions step-by-step without revealing the heuristic reasoning by which those steps were discovered.

trilinear pole and trilinear polar¶

Domain & Locus: Modern Triangle Geometry (§ 549)
Symbolic / Center Identity: Harmonic axis of a point w.r.t. a triangle
Edition Loci: f. 0263 (p. 244)

Given a point P not on the sides of ABC, the intersections of its cevians AP, BP, CP with the opposite sides yield harmonic conjugates whose three points are collinear on the trilinear polar of P (for P = centroid G, the line at infinity; for P = symmedian point K, the Lemoine axis).

tritangent circles (incircle and excircles)¶

Domain & Locus: Triangle Geometry (§ 1, § 118)
Symbolic / Center Identity: (I), (Ia), (Ib), (Ic); incenter I = X(1)
Edition Loci: f. 0092 (p. 73)

The four circles simultaneously tangent to all three sidelines BC, CA, AB of a triangle: the inscribed circle (incircle, center I = X(1), radius r) and the three escribed circles (excircles, centers Ia, Ib, Ic, radii ra, rb, rc).

Tucker circles and Taylor circle¶

Domain & Locus: Modern Triangle Geometry (§§ 684–689)
Symbolic / Center Identity: Generalized six-point circles centered on OK
Edition Loci: f. 0303 (p. 284)

One-parameter family of six-point circles generated by homothetic triangles inscribed with antiparallel/parallel chords, encompassing the circumcircle, first and second Lemoine circles, and the Taylor circle (§ 689).

Technical Glossary of Modern Synthetic Geometry¶