Encyclopedia of Triangle Centers

Thematic Collections (X₁–X₄₀₀₀)

X₁–X₁₉Classical & Ancient Centers (19 centers)

The foundational centers of classical triangle geometry: the incenter, centroid, circumcenter, orthocenter, nine-point center, symmedian point, Gergonne, Nagel, Mittenpunkt, Spieker, Feuerbach, Fermat, isodynamic, Napoleon, and Clawson points.

X₂₀–X₃₀Euler Line Centers (11 centers)

Classical centers lying on the Euler line X₂ X₃, including the De Longchamps point, Schiffler point, Exeter point, far-out point, homothetic center of the orthic and tangential triangles, and the Euler infinity point.

X₃₁–X₅₈Power Points, Circle Similicenters & Named Centers (28 centers)

2nd and 3rd power points, the Bevan point, Brocard midpoint, Kosnita point, and internal and external centers of similitude of the circumcircle, incircle, and nine-point circle.

X₅₉–X₉₇Isogonal & Isotomic Conjugates of Classical Centers (39 centers)

Isogonal conjugates (X₅₉–X₇₄) and isotomic and cross conjugates (X₇₅–X₉₇) of the classical centers X₁–X₅₈.

X₉₈–X₁₁₂Centers on the Circumcircle (15 centers)

Notable points on the circumcircle of ABC, including the Tarry point, Steiner point, anticomplementary, Feuerbach, Kiepert, and Jerabek antipodes, and centers derived from circumcircle mappings.

X₁₁₃–X₁₃₉Centers on the Nine-Point Circle & Orthic Centers (27 centers)

Points on the nine-point circle (X₁₁₃–X₁₃₉), centers of the Kiepert, Feuerbach, and Jerabek rectangular hyperbolas, and centers transferred to the orthic triangle.

X₁₄₀–X₁₉₀Derived Triangle Centers, Isoscelizers & Ajima-Malfatti Points (51 centers)

Complements, anticomplements, and classical centers evaluated on the medial, orthic, tangential, and excentral triangles, together with congruent isoscelizer, equal detour, and Ajima-Malfatti points.

X₁₉₁–X₂₃₆Ceva Conjugates (46 centers)

Perspectors of the cevian triangle of P and the anticevian triangle of Q (P-Ceva conjugates of Q) for classical triangle centers.

X₂₃₇–X₂₄₈Line Conjugates (12 centers)

Intersections of line PQ with the trilinear polar of the isogonal conjugate of P (P-line conjugates of Q).

X₂₄₉–X₂₉₇Isogonal Conjugates II & Hirst Inverses (49 centers)

Isogonal conjugates of centers X₁₁₅–X₂₄₈, together with cross conjugates and X₂-Hirst inverses.

X₂₉₈–X₃₅₀Isotomic Conjugates II (53 centers)

Isotomic conjugates of previously cataloged triangle centers, together with X₂-Hirst inverses and radical centers.

X₃₅₁–X₄₀₀Parry Circle, Morley/Yff Centers & Pedal Centroids (50 centers)

The Parry circle center, Morley trisector centers, Yff equal-parallelians, trisected-perimeter, and equilateral-cevian points, Kenmotu point, and centroids of pedal and cevian triangles.

X₄₀₁–X₄₈₄Euler Line Substitutions & Hirst Inverses (84 centers)

Centers lying on the Euler line generated by X_k-substitutions and Hirst inversions, followed by the Tixier point, Ajima-Malfatti radical center, and Evans perspectors.

X₄₈₅–X₅₁₀Vecten Square Centers & Isoscelizer Centers (26 centers)

Inner and outer Vecten points and square-associated centers (Vierkanten in een driehoek, X₄₈₅–X₄₉₅), harmonic conjugates, and isoscelizer-associated centers (X₅₀₃–X₅₁₀).

X₅₁₁–X₅₆₅Centers on the Line at Infinity & Vector Directions (55 centers)

Points on the line at infinity x + y + z = 0 (X₅₁₁–X₅₂₆, isogonal conjugates of circumcircle points), unit vector direction centers, and classical midpoints.

X₅₆₆–X₆₄₂Brocard Axis Circles & Gibert Centers (77 centers)

Centers on the Brocard axis X₃ X₆ whose circles cut equal chords on the sidelines of ABC (X₅₆₆–X₅₈₄), congruent shrunk insquares points, and centers contributed by Bernard Gibert (X₆₁₆–X₆₄₂).

X₆₄₃–X₆₉₃Crossdifferences & Trilinear Poles (51 centers)

Trilinear multipliers for the Kiepert and Yff parabolas, crossdifferences of classical centers, and trilinear poles of central lines.

X₆₉₄–X₇₆₈Even (p, q) Circumcircle Points (75 centers)

Even trigonometric (p, q) points on the circumcircle and line at infinity, paired with their isogonal conjugates.

X₇₆₉–X₈₄₃Odd o-(p, q) Circumcircle Points (75 centers)

Odd trigonometric o-(p, q) points on the circumcircle and line at infinity, paired with their isogonal conjugates.

X₈₄₄–X₉₀₀Hatzipolakis-Yiu Centers, Hirst Inverses & Crossdifferences (57 centers)

Hatzipolakis-Yiu points, the Yiu angle point (X₈₄₈), Kiepert hyperbola multipliers, Hirst inverses, and Euler line intercepts.

X₉₀₁–X₁₀₀₀Star-Named Centers, Ehrmann Points & Anticomplements (100 centers)

Star-named centers (Agena, Achernar, Algenib, Alcor), Ehrmann points, anticomplements, and midpoints of classical triangle centers.

X₁₀₀₁–X₁₀₆₉Classical Midpoints, Euler Intercepts & Cevapoints (69 centers)

Midpoints of classical centers, Euler line intercepts, collinearity intersections, cevapoints, and pure-angle trilinear centers.

X₁₀₇₀–X₁₁₁₄Cross Conjugates, Trilinear Squares & Euler-Circumcircle Intersections (45 centers)

Cross conjugates, trilinear squares and cubes of classical centers, and the intersections of the Euler line with the circumcircle (X₁₁₁₃, X₁₁₁₄).

X₁₁₁₅–X₁₁₆₅Morley-Taylor-Marr, Hofstadter, Ehrmann & Star Centers (51 centers)

Morley-Taylor-Marr centers, Hofstadter points, the Ehrmann congruent squares point (X₁₁₄₄), Grebe triangle centers, and star-named points (Aldebaran, Algjebbah, Altair, Altais).

X₁₁₆₆–X₁₂₀₈Saragossa Points (43 centers)

1st, 2nd, and 3rd Saragossa points (Q, Q', Q″) constructed from the cevian and circumcevian triangles of classical centers.

X₁₂₀₉–X₁₂₈₅Saragossa Conjugates, Sharygin Points & Lemoine Homothetic Center (77 centers)

Isogonal and isotomic conjugates of the Saragossa points, Sharygin points, and the Lemoine homothetic center (X₁₂₈₅).

X₁₂₈₆–X₁₃₁₁Collings Transforms (26 centers)

Concurrence points on the circumcircle of the circumcircles of AB'C', A'BC', and A'B'C, where A'B'C' is the reflection of ABC in P.

X₁₃₁₂–X₁₃₅₃Moses Intersections, Circumcircle Inverses, Wernau Points & Similicenters (42 centers)

Moses orthocentroidal-Euler intersections, circumcircle inverses, 1st and 2nd Wernau points (X₁₃₃₇, X₁₃₃₈), and similicenters of the circumcircle, nine-point circle, orthocentroidal circle, and Brocard circle.

X₁₃₅₄–X₁₃₉₂Brisse Transforms, Rigby/Griffiths Points & Complementary Conjugates (39 centers)

Polynomial centers on the incircle generated by the Brisse transform (X₁₃₅₄–X₁₃₆₇), Rigby and Griffiths points, and complementary and anticomplementary conjugates.

X₁₃₉₃–X₁₄₇₇Beth Conjugates & Harmonic Conjugates (85 centers)

Fixed-point P-beth conjugates of P and harmonic conjugates along central lines.

X₁₄₇₈–X₁₅₁₁Johnson-Yff Circle, Perspectors & Star-Named Centers (34 centers)

The Johnson-Yff circle center, tangential-of-tangential perspectors, star-named centers (Bellatrix, Betelgeuse, Biham), and Napoleon and Fermat crosssums.

X₁₅₁₂–X₁₆₀₀Orthojoins, Moses Circle Centers & Star-Named Centers (89 centers)

Orthojoins H(X) — orthopoles of the trilinear polars of isogonal conjugates (X₁₅₁₂–X₁₅₆₈) — together with Moses circle similicenters and star-named centers (Capella, Caph, Cebalrai, Celaeno, Chertan, Chort).

X₁₆₀₁–X₁₆₆₁TCC-Perspectors & Tripolar Centroids (61 centers)

Perspectors of the tangential triangle and the circumcevian triangle of P (X₁₆₀₁–X₁₆₃₄), tripolar centroids (X₁₆₃₅–X₁₆₅₁), and Evans and Grinberg perspectors.

X₁₆₆₂–X₁₇₀₆Circle-Related Points (45 centers)

Internal and external centers of similitude among the circumcircle, incircle, nine-point circle, Brocard circle, 1st and 2nd Lemoine circles, Apollonius circle, and Bevan circle.

X₁₇₀₇–X₁₇₈₈Mimosa Transforms (82 centers)

Centers generated by the Mimosa transform M(X) of classical triangle centers.

X₁₇₈₉–X₁₈₂₃Inverse Mimosa Transforms & Aleph Conjugates (35 centers)

Inverse Mimosa transforms of classical triangle centers and aleph conjugates.

X₁₈₂₄–X₁₉₀₇Zosma Transforms (84 centers)

Isogonal conjugates of the inverse Mimosa transforms (Z(X)) of classical triangle centers.

X₁₉₀₈–X₂₀₀₀Bicentric Pairs PU(k) & Orthocorrespondents (93 centers)

Midpoints, crosssums, crossdifferences, trilinear poles, and trilinear products of bicentric pairs PU(k) (X₁₉₀₈–X₁₉₈₂), Vega transforms, and orthocorrespondents (X₁₉₉₂–X₂₀₀₀).

X₂₀₀₁–X₂₀₅₄Orthocorrespondents, Gallatly Circle & Euler-Vecten-Gibert Points (54 centers)

Orthocorrespondents of classical centers (X₂₀₀₁–X₂₀₀₆), similicenters and intersections of the Gallatly circle (X₂₀₀₇–X₂₀₄₀), and the six Euler-Vecten-Gibert points and harmonic conjugates (X₂₀₄₁–X₂₀₅₄).

X₂₀₅₅–X₂₁₁₉Orion Transforms, Circumcircle Inverses, PK/NK Transforms, Reflections & Kepler Points (65 centers)

Orion transforms OT(X) (X₂₀₅₅–X₂₀₆₅), Caroli and Cursa points, circumcircle inverses (X₂₀₇₀–X₂₀₈₀), PK and NK transforms (X₂₀₈₁–X₂₀₉₂), classical reflections (X₂₀₉₃–X₂₁₀₅), and Kepler points and their isotomic conjugates (X₂₁₀₆–X₂₁₁₉).

X₂₁₂₀–X₂₁₄₇Eigencenters & Eigentransforms (28 centers)

Eigentransforms ET(X) — eigencenters of the cevian triangles of classical centers — paired with their isogonal conjugates on the self-isogonal cubic Z(X) (X₂₁₂₀–X₂₁₄₃), followed by X₂-eigentransforms (X₂₁₄₄–X₂₁₄₇).

X₂₁₄₈–X₂₂₅₄Isotomic Conjugates (X₂-Isoconjugates) I (107 centers)

Isotomic conjugates (X₂-isoconjugates) of triangle centers X₅ through X₈₉₂.

X₂₂₅₅–X₂₃₆₄Isotomic Conjugates (X₂-Isoconjugates) II (110 centers)

Isotomic conjugates (X₂-isoconjugates) of triangle centers X₉₀₁ through X₂₀₉₉.

X₂₃₆₅–X₂₄₄₅Circumcircle CIR(X) Points & Gibert-Simson Transforms (81 centers)

Circumcevian-eigentransform points CIR(X) on the circumcircle and their isogonal conjugates (X₂₃₆₅–X₂₃₉₃), followed by Gibert-Simson transforms GS(X) on the Simson cubic K010 (X₂₃₉₄–X₂₄₁₉) and their isogonal conjugates on K162 (X₂₄₂₀–X₂₄₄₅).

X₂₄₄₆–X₂₅₂₀Circle & Conic Intersections, Lemoine Inverses & Radical Centers I (75 centers)

Intersections of central lines with the incircle, Bevan circle, Steiner inellipse, and Steiner circumellipse, inverses in the 1st and 2nd Lemoine circles, and circle radical centers (X₂₄₄₆–X₂₅₂₀).

X₂₅₂₁–X₂₅₉₃Circle Similicenters, Radical Centers & Isoconjugates II (73 centers)

Similicenters and intersections involving the orthocentroidal, 2nd Brocard, Parry, and Gallatly circles (X₂₅₂₁–X₂₅₇₃), followed by isogonal and X_k-isoconjugates (X₂₅₇₄–X₂₅₉₃).

X₂₅₉₄–X₂₆₇₆Bicentric Pairs PU(68)–PU(85) & Golden Arbelos Points (83 centers)

Sums, differences, crosssums, crossdifferences, trilinear poles, and midpoints of bicentric pairs PU(68) through PU(85) (X₂₅₉₄–X₂₆₇₀), together with the 1st and 2nd Golden Arbelos points (X₂₆₇₁–X₂₆₇₆).

X₂₆₇₇–X₂₇₇₀Rigby-Simson RS(P, U) & Simson-Rigby SR(P, U) Points (94 centers)

Rigby-Simson orthopoles RS(P, U) where the Simson lines of circumcircle points P and U meet (X₂₆₇₇–X₂₆₈₆), and Simson-Rigby circumcircle points SR(P, U) whose Simson lines pass through RS(P, U) perpendicular to PU (X₂₆₈₇–X₂₇₇₀).

X₂₇₇₁–X₂₈₅₄Isogonal Conjugates of Simson-Rigby SR(P, U) Points (84 centers)

Points on the line at infinity formed as the isogonal conjugates of the Simson-Rigby circumcircle points X₂₆₈₇–X₂₇₇₀.

X₂₈₅₅–X₂₈₈₂Simson-Moses SM(P, U) Points & Conjugates (28 centers)

Simson-Moses circumcircle points SM(P, U) generated by isotomic conjugation of circumcircle pairs (X₂₈₅₅–X₂₈₆₈) and their isogonal conjugates on the line at infinity (X₂₈₆₉–X₂₈₈₂).

X₂₈₈₃–X₂₉₆₆Isogonal Conjugates with Respect to Special Triangles (84 centers)

Complementary (medial-isogonal), anticomplementary-isogonal, tangential-isogonal, and excentral-isogonal conjugates of classical triangle centers, together with trilinear and barycentric products.

X₂₉₆₇–X₃₀₀₀MacBeath Points, Dual Centers DC(X) / CD(X) & Central Line Intersections (34 centers)

The eight MacBeath points (X₂₉₆₇–X₂₉₇₄), circle radical centers (X₂₉₇₅–X₂₉₇₈), dual triangle mappings DC(X) and CD(X) (X₂₉₇₉–X₂₉₉₈), the Danneels-Bevan homothetic center (X₂₉₉₉), and the Soddy-antiorthic intersection (X₃₀₀₀).

X₃₀₀₁–X₃₁₁₈Central Line Intersections, Stevanović Points & T/G Mappings (118 centers)

Pairwise intersections of the Brocard, de Longchamps, Gergonne, Lemoine, Nagel, orthic, Soddy, and Fermat axes (X₃₀₀₁–X₃₀₁₉), the 4th–10th Stevanović points (X₃₀₂₀–X₃₀₂₆), antipodal circle mappings and complements (X₃₀₂₇–X₃₁₁₁), and Quang Tuan Bui’s TgX and GtX mappings (X₃₁₁₂–X₃₁₁₈).

X₃₁₁₉–X₃₂₁₇Danneels Perspectors & X_i-Ceva Conjugates (99 centers)

Eric Danneels’ parallel-cevian perspectors (X₃₁₁₉–X₃₁₂₆) and a systematic catalog of X_i-Ceva conjugates of classical and central points (X₃₁₂₇–X₃₂₁₇).

X₃₂₁₈–X₃₃₀₉Side-Swapping SS(a → bc) Points, Parabola Vertices & Morley Equilateral Triangles (92 centers)

Side-swapping transformations SS(a → bc), vertices of the inscribed Kiepert and Yff parabolas (X₃₂₃₃–X₃₂₃₄), reflections in X₁ (X₃₂₄₁–X₃₂₄₅), and centers and perspectors of the six equilateral triangles related to the 1st, 2nd, and 3rd Morley triangles (X₃₂₇₂–X₃₂₈₃).

X₃₃₁₀–X₃₄₁₄Incircle Transforms T(U), Trigonometric Offset Points & Kiepert Cusps (105 centers)

Barycentric products of perpendicular directions (X₃₃₁₀), incircle transforms T(U) lying on the incircle (X₃₃₁₈–X₃₃₂₈), Darboux-cubic reflections and conjugates (X₃₃₄₁–X₃₃₅₈), trigonometric phase-shifted centers cos/sin/sec/csc/tan/cot(A ± θ) (X₃₃₆₄–X₃₄₀₉), and the Brocard-Kiepert-Fermat cusps (X₃₄₁₁–X₃₄₁₄).

X₃₄₁₅–X₃₅₁₄Vertex Conjugates, Saragossa Points & Neuberg Cubic Intersections (100 centers)

U-vertex conjugates of X formed from circumcevian vertex triangles (X₃₄₁₅–X₃₄₅₈), 1st Saragossa points (X₃₄₄₉–X₃₄₅₄), X₅- and X₃₀-Ceva conjugates on the Neuberg cubic (X₃₄₆₀–X₃₄₈₈), and the 1st dilation center and its circumcircle inverse (X₃₅₁₃–X₃₅₁₄).

X₃₅₁₅–X₃₆₀₁Euler Line Intercepts & Named Geometric Centers (87 centers)

Euler line intercepts of central lines (X₃₅₁₅–X₃₅₅₂), together with the Dosa, Pappus, Moses circumcircle/infinity, Don Quixote, Beltrami-Euler, Beltrami-Parry, Myakishev conic, Dixit, Odehnal, and Liu points (X₃₅₅₅–X₃₆₀₁).

X₃₆₀₂–X₃₆₆₂Morley Cubics, Medial Homothetic Centers H(X; M, t) & Kirikami-Schiffler Points (61 centers)

Centers associated with the three Morley cubics K029–K031 (X₃₆₀₂–X₃₆₁₅), medial-triangle t-dilation homothetic centers H(X; M, t) and their isogonal conjugates H⁻¹(X; M, t) (X₃₆₁₆–X₃₆₃₇), Soddy-Gergonne, Kirikami, Alkaid, and Almach points (X₃₆₃₈–X₃₆₄₆), and Kirikami-Schiffler points KS(T) (X₃₆₄₇–X₃₆₆₂).

X₃₆₆₃–X₃₇₅₇Central Triangle Combos Xcom(T) I: Intouch, Extouch, Incentral & Hexyl Triangles (95 centers)

Combo centers X_k-com(T) evaluated on the intouch (X₃₆₆₃–X₃₆₇₇), extouch (X₃₆₇₈–X₃₇₁₉), incentral (X₃₇₂₀–X₃₇₂₈), excentral, tangential, 1st Brocard, and hexyl triangles (X₃₇₂₉–X₃₇₃₈), plus matrix-product combos n(incentral)*n(medial) and inverse-triangle combos (X₃₇₃₉–X₃₇₅₇).

X₃₇₅₈–X₃₈₆₁Central Triangle Combos Xcom(T) II: Parameterized T(f, g) & Euler Triangles (104 centers)

Combo centers X_k-com[T(f, g)] and X_k-com[Inverse(nT(f, g))] for parameterized central triangles (X₃₇₅₈–X₃₈₁₀), followed by combo centers on the 1st through 5th Euler triangles (X₃₈₁₁–X₃₈₆₁).

X₃₈₆₂–X₄₀₀₀Inverse Central Triangle Combos Xcom[Inverse(n(T))] (139 centers)

Perspectors of ABC and parameterized central triangles T(f, g) (X₃₈₆₂–X₃₈₆₇), inverse 3rd/4th/5th Euler triangle combos (X₃₈₆₈–X₃₉₁₀), and combos and inverse-triangle combos across the intouch, extouch, orthic, tangential, hexyl, Brocard, incentral, excentral, intangents, and extangents triangles (X₃₉₁₁–X₄₀₀₀).