Content of type 'collection'

X_1–X_19 — Classical & Ancient Centers

X_1–X_19

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.
X_20–X_30 — Euler Line Centers

X_20–X_30

Classical centers lying on the Euler line X_2 X_3, 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.
X_31–X_58 — Power Points, Circle Similicenters & Named Centers

X_31–X_58

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.
X_59–X_97 — Isogonal & Isotomic Conjugates of Classical Centers

X_59–X_97

Isogonal conjugates (X_59–X_74) and isotomic and cross conjugates (X_75–X_97) of the classical centers X_1–X_58.
X_98–X_112 — Centers on the Circumcircle

X_98–X_112

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.
X_113–X_139 — Centers on the Nine-Point Circle & Orthic Centers

X_113–X_139

Points on the nine-point circle (X_113–X_139), centers of the Kiepert, Feuerbach, and Jerabek rectangular hyperbolas, and centers transferred to the orthic triangle.
X_140–X_190 — Derived Triangle Centers, Isoscelizers & Ajima-Malfatti Points

X_140–X_190

Complements, anticomplements, and classical centers evaluated on the medial, orthic, tangential, and excentral triangles, together with congruent isoscelizer, equal detour, and Ajima-Malfatti points.
X_191–X_236 — Ceva Conjugates

X_191–X_236

Perspectors of the cevian triangle of P and the anticevian triangle of Q (P-Ceva conjugates of Q) for classical triangle centers.
X_237–X_248 — Line Conjugates

X_237–X_248

Intersections of line PQ with the trilinear polar of the isogonal conjugate of P (P-line conjugates of Q).
X_249–X_297 — Isogonal Conjugates II & Hirst Inverses

X_249–X_297

Isogonal conjugates of centers X_115–X_248, together with cross conjugates and X_2-Hirst inverses.
X_298–X_350 — Isotomic Conjugates II

X_298–X_350

Isotomic conjugates of previously cataloged triangle centers, together with X_2-Hirst inverses and radical centers.
X_351–X_400 — Parry Circle, Morley/Yff Centers & Pedal Centroids

X_351–X_400

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.
X_401–X_484 — Euler Line Substitutions & Hirst Inverses

X_401–X_484

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.
X_485–X_510 — Vecten Square Centers & Isoscelizer Centers

X_485–X_510

Inner and outer Vecten points and square-associated centers (Vierkanten in een driehoek, X_485–X_495), harmonic conjugates, and isoscelizer-associated centers (X_503–X_510).
X_511–X_565 — Centers on the Line at Infinity & Vector Directions

X_511–X_565

Points on the line at infinity x + y + z = 0 (X_511–X_526, isogonal conjugates of circumcircle points), unit vector direction centers, and classical midpoints.
X_566–X_642 — Brocard Axis Circles & Gibert Centers

X_566–X_642

Centers on the Brocard axis X_3 X_6 whose circles cut equal chords on the sidelines of ABC (X_566–X_584), congruent shrunk insquares points, and centers contributed by Bernard Gibert (X_616–X_642).
X_643–X_693 — Crossdifferences & Trilinear Poles

X_643–X_693

Trilinear multipliers for the Kiepert and Yff parabolas, crossdifferences of classical centers, and trilinear poles of central lines.
X_694–X_768 — Even (p, q) Circumcircle Points

X_694–X_768

Even trigonometric (p, q) points on the circumcircle and line at infinity, paired with their isogonal conjugates.
X_769–X_843 — Odd o-(p, q) Circumcircle Points

X_769–X_843

Odd trigonometric o-(p, q) points on the circumcircle and line at infinity, paired with their isogonal conjugates.
X_844–X_900 — Hatzipolakis-Yiu Centers, Hirst Inverses & Crossdifferences

X_844–X_900

Hatzipolakis-Yiu points, the Yiu angle point (X_848), Kiepert hyperbola multipliers, Hirst inverses, and Euler line intercepts.
X_901–X_1000 — Star-Named Centers, Ehrmann Points & Anticomplements

X_901–X_1000

Star-named centers (Agena, Achernar, Algenib, Alcor), Ehrmann points, anticomplements, and midpoints of classical triangle centers.
X_1001–X_1069 — Classical Midpoints, Euler Intercepts & Cevapoints

X_1001–X_1069

Midpoints of classical centers, Euler line intercepts, collinearity intersections, cevapoints, and pure-angle trilinear centers.
X_1070–X_1114 — Cross Conjugates, Trilinear Squares & Euler-Circumcircle Intersections

X_1070–X_1114

Cross conjugates, trilinear squares and cubes of classical centers, and the intersections of the Euler line with the circumcircle (X_1113, X_1114).
X_1115–X_1165 — Morley-Taylor-Marr, Hofstadter, Ehrmann & Star Centers

X_1115–X_1165

Morley-Taylor-Marr centers, Hofstadter points, the Ehrmann congruent squares point (X_1144), Grebe triangle centers, and star-named points (Aldebaran, Algjebbah, Altair, Altais).
X_1166–X_1208 — Saragossa Points

X_1166–X_1208

1st, 2nd, and 3rd Saragossa points (Q, Q', Q″) constructed from the cevian and circumcevian triangles of classical centers.
X_1209–X_1285 — Saragossa Conjugates, Sharygin Points & Lemoine Homothetic Center

X_1209–X_1285

Isogonal and isotomic conjugates of the Saragossa points, Sharygin points, and the Lemoine homothetic center (X_1285).
X_1286–X_1311 — Collings Transforms

X_1286–X_1311

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.
X_1312–X_1353 — Moses Intersections, Circumcircle Inverses, Wernau Points & Similicenters

X_1312–X_1353

Moses orthocentroidal-Euler intersections, circumcircle inverses, 1st and 2nd Wernau points (X_1337, X_1338), and similicenters of the circumcircle, nine-point circle, orthocentroidal circle, and Brocard circle.
X_1354–X_1392 — Brisse Transforms, Rigby/Griffiths Points & Complementary Conjugates

X_1354–X_1392

Polynomial centers on the incircle generated by the Brisse transform (X_1354–X_1367), Rigby and Griffiths points, and complementary and anticomplementary conjugates.
X_1393–X_1477 — Beth Conjugates & Harmonic Conjugates

X_1393–X_1477

Fixed-point P-beth conjugates of P and harmonic conjugates along central lines.
X_1478–X_1511 — Johnson-Yff Circle, Perspectors & Star-Named Centers

X_1478–X_1511

The Johnson-Yff circle center, tangential-of-tangential perspectors, star-named centers (Bellatrix, Betelgeuse, Biham), and Napoleon and Fermat crosssums.
X_1512–X_1600 — Orthojoins, Moses Circle Centers & Star-Named Centers

X_1512–X_1600

Orthojoins H(X) — orthopoles of the trilinear polars of isogonal conjugates (X_1512–X_1568) — together with Moses circle similicenters and star-named centers (Capella, Caph, Cebalrai, Celaeno, Chertan, Chort).
X_1601–X_1661 — TCC-Perspectors & Tripolar Centroids

X_1601–X_1661

Perspectors of the tangential triangle and the circumcevian triangle of P (X_1601–X_1634), tripolar centroids (X_1635–X_1651), and Evans and Grinberg perspectors.
X_1662–X_1706 — Circle-Related Points

X_1662–X_1706

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.
X_1707–X_1788 — Mimosa Transforms

X_1707–X_1788

Centers generated by the Mimosa transform M(X) of classical triangle centers.
X_1789–X_1823 — Inverse Mimosa Transforms & Aleph Conjugates

X_1789–X_1823

Inverse Mimosa transforms of classical triangle centers and aleph conjugates.
X_1824–X_1907 — Zosma Transforms

X_1824–X_1907

Isogonal conjugates of the inverse Mimosa transforms (Z(X)) of classical triangle centers.
X_1908–X_2000 — Bicentric Pairs PU(k) & Orthocorrespondents

X_1908–X_2000

Midpoints, crosssums, crossdifferences, trilinear poles, and trilinear products of bicentric pairs PU(k) (X_1908–X_1982), Vega transforms, and orthocorrespondents (X_1992–X_2000).
X_2001–X_2054 — Orthocorrespondents, Gallatly Circle & Euler-Vecten-Gibert Points

X_2001–X_2054

Orthocorrespondents of classical centers (X_2001–X_2006), similicenters and intersections of the Gallatly circle (X_2007–X_2040), and the six Euler-Vecten-Gibert points and harmonic conjugates (X_2041–X_2054).
X_2055–X_2119 — Orion Transforms, Circumcircle Inverses, PK/NK Transforms, Reflections & Kepler Points

X_2055–X_2119

Orion transforms OT(X) (X_2055–X_2065), Caroli and Cursa points, circumcircle inverses (X_2070–X_2080), PK and NK transforms (X_2081–X_2092), classical reflections (X_2093–X_2105), and Kepler points and their isotomic conjugates (X_2106–X_2119).
X_2120–X_2147 — Eigencenters & Eigentransforms

X_2120–X_2147

Eigentransforms ET(X) — eigencenters of the cevian triangles of classical centers — paired with their isogonal conjugates on the self-isogonal cubic Z(X) (X_2120–X_2143), followed by X_2-eigentransforms (X_2144–X_2147).
X_2148–X_2254 — Isotomic Conjugates (X_2-Isoconjugates) I

X_2148–X_2254

Isotomic conjugates (X_2-isoconjugates) of triangle centers X_5 through X_892.
X_2255–X_2364 — Isotomic Conjugates (X_2-Isoconjugates) II

X_2255–X_2364

Isotomic conjugates (X_2-isoconjugates) of triangle centers X_901 through X_2099.
X_2365–X_2445 — Circumcircle CIR(X) Points & Gibert-Simson Transforms

X_2365–X_2445

Circumcevian-eigentransform points CIR(X) on the circumcircle and their isogonal conjugates (X_2365–X_2393), followed by Gibert-Simson transforms GS(X) on the Simson cubic K010 (X_2394–X_2419) and their isogonal conjugates on K162 (X_2420–X_2445).
X_2446–X_2520 — Circle & Conic Intersections, Lemoine Inverses & Radical Centers I

X_2446–X_2520

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 (X_2446–X_2520).
X_2521–X_2593 — Circle Similicenters, Radical Centers & Isoconjugates II

X_2521–X_2593

Similicenters and intersections involving the orthocentroidal, 2nd Brocard, Parry, and Gallatly circles (X_2521–X_2573), followed by isogonal and X_k-isoconjugates (X_2574–X_2593).
X_2594–X_2676 — Bicentric Pairs PU(68)–PU(85) & Golden Arbelos Points

X_2594–X_2676

Sums, differences, crosssums, crossdifferences, trilinear poles, and midpoints of bicentric pairs PU(68) through PU(85) (X_2594–X_2670), together with the 1st and 2nd Golden Arbelos points (X_2671–X_2676).
X_2677–X_2770 — Rigby-Simson RS(P, U) & Simson-Rigby SR(P, U) Points

X_2677–X_2770

Rigby-Simson orthopoles RS(P, U) where the Simson lines of circumcircle points P and U meet (X_2677–X_2686), and Simson-Rigby circumcircle points SR(P, U) whose Simson lines pass through RS(P, U) perpendicular to PU (X_2687–X_2770).
X_2771–X_2854 — Isogonal Conjugates of Simson-Rigby SR(P, U) Points

X_2771–X_2854

Points on the line at infinity formed as the isogonal conjugates of the Simson-Rigby circumcircle points X_2687–X_2770.
X_2855–X_2882 — Simson-Moses SM(P, U) Points & Conjugates

X_2855–X_2882

Simson-Moses circumcircle points SM(P, U) generated by isotomic conjugation of circumcircle pairs (X_2855–X_2868) and their isogonal conjugates on the line at infinity (X_2869–X_2882).
X_2883–X_2966 — Isogonal Conjugates with Respect to Special Triangles

X_2883–X_2966

Complementary (medial-isogonal), anticomplementary-isogonal, tangential-isogonal, and excentral-isogonal conjugates of classical triangle centers, together with trilinear and barycentric products.
X_2967–X_3000 — MacBeath Points, Dual Centers DC(X) / CD(X) & Central Line Intersections

X_2967–X_3000

The eight MacBeath points (X_2967–X_2974), circle radical centers (X_2975–X_2978), dual triangle mappings DC(X) and CD(X) (X_2979–X_2998), the Danneels-Bevan homothetic center (X_2999), and the Soddy-antiorthic intersection (X_3000).
X_3001–X_3118 — Central Line Intersections, Stevanović Points & T/G Mappings

X_3001–X_3118

Pairwise intersections of the Brocard, de Longchamps, Gergonne, Lemoine, Nagel, orthic, Soddy, and Fermat axes (X_3001–X_3019), the 4th–10th Stevanović points (X_3020–X_3026), antipodal circle mappings and complements (X_3027–X_3111), and Quang Tuan Bui’s TgX and GtX mappings (X_3112–X_3118).
X_3119–X_3217 — Danneels Perspectors & X_i-Ceva Conjugates

X_3119–X_3217

Eric Danneels’ parallel-cevian perspectors (X_3119–X_3126) and a systematic catalog of X_i-Ceva conjugates of classical and central points (X_3127–X_3217).
X_3218–X_3309 — Side-Swapping SS(a → bc) Points, Parabola Vertices & Morley Equilateral Triangles

X_3218–X_3309

Side-swapping transformations SS(a → bc), vertices of the inscribed Kiepert and Yff parabolas (X_3233–X_3234), reflections in X_1 (X_3241–X_3245), and centers and perspectors of the six equilateral triangles related to the 1st, 2nd, and 3rd Morley triangles (X_3272–X_3283).
X_3310–X_3414 — Incircle Transforms T(U), Trigonometric Offset Points & Kiepert Cusps

X_3310–X_3414

Barycentric products of perpendicular directions (X_3310), incircle transforms T(U) lying on the incircle (X_3318–X_3328), Darboux-cubic reflections and conjugates (X_3341–X_3358), trigonometric phase-shifted centers cos/sin/sec/csc/tan/cot(A ± θ) (X_3364–X_3409), and the Brocard-Kiepert-Fermat cusps (X_3411–X_3414).
X_3415–X_3514 — Vertex Conjugates, Saragossa Points & Neuberg Cubic Intersections

X_3415–X_3514

U-vertex conjugates of X formed from circumcevian vertex triangles (X_3415–X_3458), 1st Saragossa points (X_3449–X_3454), X_5- and X_30-Ceva conjugates on the Neuberg cubic (X_3460–X_3488), and the 1st dilation center and its circumcircle inverse (X_3513–X_3514).
X_3515–X_3601 — Euler Line Intercepts & Named Geometric Centers

X_3515–X_3601

Euler line intercepts of central lines (X_3515–X_3552), together with the Dosa, Pappus, Moses circumcircle/infinity, Don Quixote, Beltrami-Euler, Beltrami-Parry, Myakishev conic, Dixit, Odehnal, and Liu points (X_3555–X_3601).
X_3602–X_3662 — Morley Cubics, Medial Homothetic Centers H(X; M, t) & Kirikami-Schiffler Points

X_3602–X_3662

Centers associated with the three Morley cubics K029–K031 (X_3602–X_3615), medial-triangle t-dilation homothetic centers H(X; M, t) and their isogonal conjugates H⁻¹(X; M, t) (X_3616–X_3637), Soddy-Gergonne, Kirikami, Alkaid, and Almach points (X_3638–X_3646), and Kirikami-Schiffler points KS(T) (X_3647–X_3662).
X_3663–X_3757 — Central Triangle Combos Xcom(T) I: Intouch, Extouch, Incentral & Hexyl Triangles

X_3663–X_3757

Combo centers X_k-com(T) evaluated on the intouch (X_3663–X_3677), extouch (X_3678–X_3719), incentral (X_3720–X_3728), excentral, tangential, 1st Brocard, and hexyl triangles (X_3729–X_3738), plus matrix-product combos n(incentral)*n(medial) and inverse-triangle combos (X_3739–X_3757).
X_3758–X_3861 — Central Triangle Combos Xcom(T) II: Parameterized T(f, g) & Euler Triangles

X_3758–X_3861

Combo centers X_k-com[T(f, g)] and X_k-com[Inverse(nT(f, g))] for parameterized central triangles (X_3758–X_3810), followed by combo centers on the 1st through 5th Euler triangles (X_3811–X_3861).
X_3862–X_4000 — Inverse Central Triangle Combos Xcom[Inverse(n(T))]

X_3862–X_4000

Perspectors of ABC and parameterized central triangles T(f, g) (X_3862–X_3867), inverse 3rd/4th/5th Euler triangle combos (X_3868–X_3910), and combos and inverse-triangle combos across the intouch, extouch, orthic, tangential, hexyl, Brocard, incentral, excentral, intangents, and extangents triangles (X_3911–X_4000).