Besondere Punkte/Special Points

Roman Chijner, Feb 26, 2019

Dynamic Visualization of construction of particular positions of some special points. / Dynamische Visualisierung der Konstruktion der besonderen Punkte . From: List of My Public Books on GeoGebra Topics: Constructing polyhedra -[url]https://www.geogebra.org/m/eabstecp[/url]

Table of Contents

  1. Fermat-Punkt
    1. Fermat-Punkt
    2. Geometric median 2D. Eine dynamische Konstruktion des Fermat-Punktes im n-Eck mit Hilfe von der Weiszfeld- Iterationsverfahren
    3. Geometric median 3D. Eine dynamische Konstruktion des Fermat-Punktes im Polyeder mit n Ecken auf der Kugeloberfläche mit Hilfe von der Weiszfeld- Iterationsverfahren
    4. Die zum Fermat-Problem gehörende Maximumaufgabe
    5. El punt de Fermat
    6. Weiszfeld Algorithmus
    7. Fermat-Punkt, verallgemeinerte Problemstellung
    8. Generalized Fermat-Weber problem
    9. Weighted geometric median (Weber problem)
    10. Geometric median of tetrahedron
    11. Geometric median of 4 points
    12. Robot Fermat point
    13. Robot Steiner points
    14. Der Fermatsche Punkt
    15. Fermat point triangle
    16. Punto de Fermat - Demostración
    17. Problema de Weber (Mediana Geométrica)
    18. Fermat Points and the Centroid
    19. Der Neubau des Kraftwerks
    20. Der Neubau des Kraftwerks
    21. centros triángulo
    22. Fermat Point of a Triangle
    23. Punto de Fermat - Triángulo equilátero asociado
    24. Points de Fermat
    25. Point de Fermat - Steiner - Torricelli
    26. Arbre de Steiner de quatre points
    27. Point de Fermat - Torriccelli
    28. Guess the Centroid!
    29. Feuerbach circle
    30. Triunghi-I-cevian-inaltime-bisectoare
    31. Thompson Cubic
    32. Four concyclic triangle centers IV
    33. The centre of mass of a lamina between two curves.
    34. Finding the centre of mass of a lamina plane introduction.
    35. Fermat-Euler
    36. Una demostración del Teorema
    37. Triangle Properties Explorer
    38. Generalización del Teorema de la Bisectriz:
    39. Baryzentren-Kurve mit den Potenzen von a,b,c
    40. Punto de Fermat en triángulos rectángulos
    41. Morley's Trisector Theorem
    42. Centroid of a planar region and planar curve
    43. Generalization of the fourth Fermat-Dao-Nhi equilateral triangle
    44. Generalization of the third Fermat-Dao-Nhi equilateral triangle
    45. Mandart ellipse et cercle
    46. 삼각형의 무게중심의 위치벡터(The position vector of the center of gravity of the triangle)
    47. Cervian triangle and areas
    48. Feuerbach théorème
    49. Centers Associated With The Incenter
    50. 三角形の垂直二等分線に対称な点の作る三角形
    51. Euler line
    52. Teorema de Routh
    53. Окружность Ван Ламуна
    54. Parabel und drei Tangenten
    55. Triangle's centroid (first moment of area)
    56. Zyklische Punkte im Dreieck
    57. 三角形の中央＝フェルマー点
    58. ESTÁTICA - Vectores en 3D
    59. Triángulos: puntos notables.
    60. Theorem von Marion
    61. Theorem von Marion 3D
    62. 完全四角形からミケル点へ
    63. Fermat-Point CAS-Function
    64. シュタイナー点
    65. シュタイナー点（４点）
    66. 五点を結ぶシュタイナー点
    67. 六点のシュタイナー点
    68. ベクトルとシュタイナー点
    69. Fermat-Punkt mit ganzzahligen Längen
    70. Point de Steiner
    71. Feynman triangle
    72. Kepler-Dreieck
    73. center of mass/weighted average
  2. Besondere Punkte im Polygon
    1. Euler-Gerade und Feuerbach-Kreis
    2. Geometric Centroid of a Parabolic Segment
    3. Viviani's theorem
    4. Special Points of Triangles - Lesson+Exploration+Practice
    5. Théorème de Viviani (2)
    6. Center of triangle
    7. Means
    8. Schar von Bezierkurven
    9. Puzzle game-5
    10. Objetos notables asociados a un triángulo
    11. Elementos del Triángulo
    12. Puntos y rectas notables del triángulo.
    13. Steiner circum- and in- ellipse
    14. Circonferenza di Feuerbach (circonferenza di Eulero dei nove punti).
    15. Cerchio di Feuerbach e ratta di Eulero
    16. Triangle Centres and exploration
    17. Golden Mean,Music Scale, Quadratic Equation
    18. Euler (démonstrations)
    19. Examining: TriangleCenter[ A, B, C, n ] and Trilinear[...
    20. 3D Center of Masses x 6 V1.1
    21. Lagrange Points V1.1
    22. Visualize 4D 3-Body Conservation of Momentum_02
    23. Retta di Eulero e punti notevoli dei triangoli
    24. Triángulos de Napoleón
    25. Teselación basada en el Teorema de Napoleón
    26. Distancia Ortocentro-vértice = 2 distancia Circuncentro-lado
    27. Euler Line
    28. Bisección de un triángulo
    29. Retta di Eulero e Circonferenza dei nove punti
    30. Centre of mass: The leaning tower of Lire
    31. Triangle Centers & Euler Line
    32. Five orthocenters lie on a circle
    33. Tres triángulos con áreas en progresión geométrica
    34. Conjugados ciclocevianos
    35. Bimedianas y baricentro de un cuadfrilátero
    36. Triángulo de Morley
    37. Lugar geométrico baricentros triángulos área cte. inscritos en parábola
    38. Six orthocenters lie on a circle
    39. Besondere Punkte im Dreieck
    40. Anteil im Dreieck
    41. Finding the centre of mass of a uniform 2D body.
    42. Finding the centre of mass of a hollow bowl.
    43. Chaos-Fraktal im Vieleck
    44. Interessante Ortslinie im Dreieck
    45. The centre of mass of a lamina bounded by a quadratic, x = a and x = b
    46. Schwerpunkt eines Vielecks
    47. Projective Geometry: Homography, Cross-ratio, Harmonic Conjugates, Polarity
    48. The centre of gravity of a rectangular lamina.
    49. Zwei besondere Dreiecks-Punkte
    50. 6-Punkte-Kreis im Dreieck
    51. Céviennes et aires
    52. Bottema's theorem
    53. Incircle, Excircles, and the 9-Point Circle
    54. Ryan Morgan conjecture
    55. Symmediane im Sehnenviereck
    56. The centroid of a parametric curve
    57. Symmedian
    58. Points remarquables du triangle
    59. Circumradio y área de un cuadrilátero cíclico en función de los lados
    60. El triángulo órtico y la circunferencia circunscrita
    61. 6-Punkte-Kreis im Dreieck
    62. Teorema de Napoleón ( comprobación )
    63. Dreiecksverhältnisse
    64. Wallace-Simson-Gerade
    65. オイラー線と直極点楕円
    66. 内接円の接線の直極点の軌跡
    67. 外心を中心にした円の直極点の軌跡
    68. El Teorema de Kariya y la hipérbola de Feuerbach
    69. Umkreismitte ABC als Höhenschnittpunkt XYZ
    70. 9-Punkte-Kegelschnitt
    71. Newton-Gauss-Linie: Kollinearität im Dreieck
    72. 完全四角形のスタイナー円の性質（補題）
    73. 完全四角形の向かい合う辺の垂直二等分線の交点
    74. 円に内接する完全四角形のミケル点
    75. 内接する完全四角形のミケル点が垂直の証明
    76. Euler-Geraden im Sehnenviereck
    77. luoghi e punti notevoli del triangolo
  3. Studying the properties of the centroid
    1. 2D Dynamic construction for studying the properties of the geometric center.
    2. 3D Dynamic construction for studying the properties of the centroid.
    3. Sum of squares of distances
    4. Teoremas del centroide de Pappus-Guldin
    5. Center of Mass: Person on a Floating Raft
    6. Center of Mass of a System of Point Masses
    7. Berechnung des Schwerpunkts einer Fläche
    8. Der Schwerpunkt des Erde-Mond Systems
    9. Der Schwerpunkt
    10. cyclic Quadrilateral
    11. swenson Centroid of a square
  4. Learn more about Squared Distance
    1. 2D. Explanation of Invariance of the sum of squares of distances by using the Steiner theorem.
    2. 3D. Explanation of Invariance of the sum of squares of distances by using the Steiner theorem.
    3. 2D. Explanation of Invariance of the sum of squares of distances by using the Steiner theorem.
    4. triangle Gizmo
    5. development on regular sqrs
    6. developments on regular polys
    7. Symmetry numbers in irregular polygons
    8. Symmetry number & conditions in right triangles
    9. Symmetrical beauty of polygons
    10. beauty of powers-2b
    11. Sliding trigon
    12. Analogy between statics & geometry
    13. Analogy between statics & geometry
    14. Hidden symmetry in an Atom
    15. Ultimate symmetry in 3D
    16. symmetry of two points in sph. & cyldr.
    17. Ultimate symmetry in cone-sphere sys.
    18. Symmetry in sphere & dodecahedron case
    19. 1-Concentric icosahedron & sphere
    20. Concentric dodecahedron & sphere
    21. 3D correspondence of symmetry spheres-B
    22. Symmetry number for 10 points
    23. Symmetry number for 5 points
    24. General solution for symmetry numbers on surfaces-B
    25. Symmetry number of a complex system
    26. Symmetry number of a complex system
    27. symmetry number for two overlapped ellipses
    28. David star overlapped
    29. symmetry numbers & parabolic mirror
    30. Mobilis in mobili symmetry
    31. Mobilis in mobili symmetry (generalised)
    32. Plane reg. polygon in a sphere
    33. Symmetry in random movables
    34. Beautiful regular pentagon-B
    35. Beautiful regular pentagon-A
    36. 4 - Tetrahedron
    37. 3 - Octahedron
    38. 2 - Icosahedron
    39. 1 - Dodecahedron
    40. 0 - Definition of platonic sets
    41. Plane equilateral triang. & sphere
    42. Regular octagon & sphere
    43. Symmetry number for double octagons-A
    44. Symmetry number for double octagons-B
    45. Three squares in a screen
    46. Three squares in a screen overlapped
    47. Definition of rules for symmetry
    48. Definition of rules for symmetry-B
    49. Password generator-A
    50. Password generator-B
    51. Extremely high powers
    52. Exremely high powers
    53. Extremely high powers
    54. Comments on extremely high powers
    55. Difference of extremely high powers-5
    56. An objection to transcendental definition
    57. New theorem of Pythagoras
    58. new theorem of pythagoras for circumferences
    59. new theorem of pythagoras for non-similar trigons
    60. New theorem of pythagoras for non-similar circles
    61. Game of delicate balance
    62. Puzzle-4
    63. Puzzle-2
    64. Ponzi game
    65. Symmetry of peripheral circles
    66. diplomatic relations
    67. Cable-car functions
    68. Four birds on a line
    69. Proxima Centauri
    70. Hexagons around hexagon
    71. Canis Major
    72. Canis Major-A
    73. Sagittarius-3D
    74. Sagittarius 3D generalized
    75. Andromeda
    76. imami azam EBU HANIFE
    77. Piscis Austrinus
    78. Pegasus
    79. Cygnus
    80. Suma de distancias al cuadrado constante
    81. Suma de cuadrados constante
    82. Quadratsumme der Radialstrecken
    83. Summe der 4.Potenz der Diagonalen
    84. Fünfeck mit Kreis (Baryzentrum)
  5. Symmedians
    1. X(85) Isogonal conjugate of X(41)
    2. Isogonal Conjugates Proof of Existence (Symmedians)
    3. Isogonal Conjugate of line through Centroid
    4. The Kiepert Hyperbola and the Brocard Axis
    5. Symmedian Point

1.

Fermat-Punkt

  • 1. Fermat-Punkt
  • 2. Geometric median 2D. Eine dynamische Konstruktion des Fermat-Punktes im n-Eck mit Hilfe von der Weiszfeld- Iterationsverfahren
  • 3. Geometric median 3D. Eine dynamische Konstruktion des Fermat-Punktes im Polyeder mit n Ecken auf der Kugeloberfläche mit Hilfe von der Weiszfeld- Iterationsverfahren
  • 4. Die zum Fermat-Problem gehörende Maximumaufgabe
  • 5. El punt de Fermat
  • 6. Weiszfeld Algorithmus
  • 7. Fermat-Punkt, verallgemeinerte Problemstellung
  • 8. Generalized Fermat-Weber problem
  • 9. Weighted geometric median (Weber problem)
  • 10. Geometric median of tetrahedron
  • 11. Geometric median of 4 points
  • 12. Robot Fermat point
  • 13. Robot Steiner points
  • 14. Der Fermatsche Punkt
  • 15. Fermat point triangle
  • 16. Punto de Fermat - Demostración
  • 17. Problema de Weber (Mediana Geométrica)
  • 18. Fermat Points and the Centroid
  • 19. Der Neubau des Kraftwerks
  • 20. Der Neubau des Kraftwerks
  • 21. centros triángulo
  • 22. Fermat Point of a Triangle
  • 23. Punto de Fermat - Triángulo equilátero asociado
  • 24. Points de Fermat
  • 25. Point de Fermat - Steiner - Torricelli
  • 26. Arbre de Steiner de quatre points
  • 27. Point de Fermat - Torriccelli
  • 28. Guess the Centroid!
  • 29. Feuerbach circle
  • 30. Triunghi-I-cevian-inaltime-bisectoare
  • 31. Thompson Cubic
  • 32. Four concyclic triangle centers IV
  • 33. The centre of mass of a lamina between two curves.
  • 34. Finding the centre of mass of a lamina plane introduction.
  • 35. Fermat-Euler
  • 36. Una demostración del Teorema
  • 37. Triangle Properties Explorer
  • 38. Generalización del Teorema de la Bisectriz:
  • 39. Baryzentren-Kurve mit den Potenzen von a,b,c
  • 40. Punto de Fermat en triángulos rectángulos
  • 41. Morley's Trisector Theorem
  • 42. Centroid of a planar region and planar curve
  • 43. Generalization of the fourth Fermat-Dao-Nhi equilateral triangle
  • 44. Generalization of the third Fermat-Dao-Nhi equilateral triangle
  • 45. Mandart ellipse et cercle
  • 46. 삼각형의 무게중심의 위치벡터(The position vector of the center of gravity of the triangle)
  • 47. Cervian triangle and areas
  • 48. Feuerbach théorème
  • 49. Centers Associated With The Incenter
  • 50. 三角形の垂直二等分線に対称な点の作る三角形
  • 51. Euler line
  • 52. Teorema de Routh
  • 53. Окружность Ван Ламуна
  • 54. Parabel und drei Tangenten
  • 55. Triangle's centroid (first moment of area)
  • 56. Zyklische Punkte im Dreieck
  • 57. 三角形の中央＝フェルマー点
  • 58. ESTÁTICA - Vectores en 3D
  • 59. Triángulos: puntos notables.
  • 60. Theorem von Marion
  • 61. Theorem von Marion 3D
  • 62. 完全四角形からミケル点へ
  • 63. Fermat-Point CAS-Function
  • 64. シュタイナー点
  • 65. シュタイナー点（４点）
  • 66. 五点を結ぶシュタイナー点
  • 67. 六点のシュタイナー点
  • 68. ベクトルとシュタイナー点
  • 69. Fermat-Punkt mit ganzzahligen Längen
  • 70. Point de Steiner
  • 71. Feynman triangle
  • 72. Kepler-Dreieck
  • 73. center of mass/weighted average

1.1.

Fermat-Punkt

[color=#000000][b] Gibt es in jedem Dreieck einen Punkt F so, daß die Summe der Entfernungen von F zu den drei Eckpunkten minimal ist?[br] [/b]Diese Problemstellung taucht zum ersten Mal um die Mitte des 17. Jahrhunderts auf. Urheber ist der französischer Mathematiker, Jurist und Parlamentsrat [url=http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fermat.html]Pierre de Fermat [/url] (1601-1665). [br][color=#444444][color=#000000][u]Konstruktion des Fermat-Punkts:[/u][/color][/color][br]● Über den Seiten eines beliebigen Dreiecks ([i]Innenwinkel ≤ 120°[/i]) werden gleichseitige Dreiecke errichtet und deren freie Eckpunkte mit den gegenüberliegenden Eckpunkten des Ausgangsdreiecks verbunden.[/color][br][color=#000000]● Wenn ein Dreieck einen [i]Winkel[/i] von [i]mehr als 120 °[/i] hat, befindet sich der Fermat-Punkt am stumpfen Winkel.[/color][br]
Bernat Link:[url=https://www.geogebra.org/material/show/id/wksuzsqc] https://www.geogebra.org/material/show/id/wksuzsqc[/url][br]zu einem sehr schönen Applet mit einer Illustration auf einer zweidimensionalen Oberfläche
– Roman Chijner

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2.

Besondere Punkte im Polygon

  • 1. Euler-Gerade und Feuerbach-Kreis
  • 2. Geometric Centroid of a Parabolic Segment
  • 3. Viviani's theorem
  • 4. Special Points of Triangles - Lesson+Exploration+Practice
  • 5. Théorème de Viviani (2)
  • 6. Center of triangle
  • 7. Means
  • 8. Schar von Bezierkurven
  • 9. Puzzle game-5
  • 10. Objetos notables asociados a un triángulo
  • 11. Elementos del Triángulo
  • 12. Puntos y rectas notables del triángulo.
  • 13. Steiner circum- and in- ellipse
  • 14. Circonferenza di Feuerbach (circonferenza di Eulero dei nove punti).
  • 15. Cerchio di Feuerbach e ratta di Eulero
  • 16. Triangle Centres and exploration
  • 17. Golden Mean,Music Scale, Quadratic Equation
  • 18. Euler (démonstrations)
  • 19. Examining: TriangleCenter[ A, B, C, n ] and Trilinear[...
  • 20. 3D Center of Masses x 6 V1.1
  • 21. Lagrange Points V1.1
  • 22. Visualize 4D 3-Body Conservation of Momentum_02
  • 23. Retta di Eulero e punti notevoli dei triangoli
  • 24. Triángulos de Napoleón
  • 25. Teselación basada en el Teorema de Napoleón
  • 26. Distancia Ortocentro-vértice = 2 distancia Circuncentro-lado
  • 27. Euler Line
  • 28. Bisección de un triángulo
  • 29. Retta di Eulero e Circonferenza dei nove punti
  • 30. Centre of mass: The leaning tower of Lire
  • 31. Triangle Centers & Euler Line
  • 32. Five orthocenters lie on a circle
  • 33. Tres triángulos con áreas en progresión geométrica
  • 34. Conjugados ciclocevianos
  • 35. Bimedianas y baricentro de un cuadfrilátero
  • 36. Triángulo de Morley
  • 37. Lugar geométrico baricentros triángulos área cte. inscritos en parábola
  • 38. Six orthocenters lie on a circle
  • 39. Besondere Punkte im Dreieck
  • 40. Anteil im Dreieck
  • 41. Finding the centre of mass of a uniform 2D body.
  • 42. Finding the centre of mass of a hollow bowl.
  • 43. Chaos-Fraktal im Vieleck
  • 44. Interessante Ortslinie im Dreieck
  • 45. The centre of mass of a lamina bounded by a quadratic, x = a and x = b
  • 46. Schwerpunkt eines Vielecks
  • 47. Projective Geometry: Homography, Cross-ratio, Harmonic Conjugates, Polarity
  • 48. The centre of gravity of a rectangular lamina.
  • 49. Zwei besondere Dreiecks-Punkte
  • 50. 6-Punkte-Kreis im Dreieck
  • 51. Céviennes et aires
  • 52. Bottema's theorem
  • 53. Incircle, Excircles, and the 9-Point Circle
  • 54. Ryan Morgan conjecture
  • 55. Symmediane im Sehnenviereck
  • 56. The centroid of a parametric curve
  • 57. Symmedian
  • 58. Points remarquables du triangle
  • 59. Circumradio y área de un cuadrilátero cíclico en función de los lados
  • 60. El triángulo órtico y la circunferencia circunscrita
  • 61. 6-Punkte-Kreis im Dreieck
  • 62. Teorema de Napoleón ( comprobación )
  • 63. Dreiecksverhältnisse
  • 64. Wallace-Simson-Gerade
  • 65. オイラー線と直極点楕円
  • 66. 内接円の接線の直極点の軌跡
  • 67. 外心を中心にした円の直極点の軌跡
  • 68. El Teorema de Kariya y la hipérbola de Feuerbach
  • 69. Umkreismitte ABC als Höhenschnittpunkt XYZ
  • 70. 9-Punkte-Kegelschnitt
  • 71. Newton-Gauss-Linie: Kollinearität im Dreieck
  • 72. 完全四角形のスタイナー円の性質（補題）
  • 73. 完全四角形の向かい合う辺の垂直二等分線の交点
  • 74. 円に内接する完全四角形のミケル点
  • 75. 内接する完全四角形のミケル点が垂直の証明
  • 76. Euler-Geraden im Sehnenviereck
  • 77. luoghi e punti notevoli del triangolo

2.1.

Euler-Gerade und Feuerbach-Kreis

eine dynamische Konstruktion
– Roman Chijner

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3.

Studying the properties of the centroid

  • 1. 2D Dynamic construction for studying the properties of the geometric center.
  • 2. 3D Dynamic construction for studying the properties of the centroid.
  • 3. Sum of squares of distances
  • 4. Teoremas del centroide de Pappus-Guldin
  • 5. Center of Mass: Person on a Floating Raft
  • 6. Center of Mass of a System of Point Masses
  • 7. Berechnung des Schwerpunkts einer Fläche
  • 8. Der Schwerpunkt des Erde-Mond Systems
  • 9. Der Schwerpunkt
  • 10. cyclic Quadrilateral
  • 11. swenson Centroid of a square

3.1.

2D Dynamic construction for studying the properties of the geometric center.

This applet presents a dynamic construction for studying the properties of the [color=#ff00ff]centroid[/color] or  [color=#ff00ff]geometric center[/color] [url=https://en.wikipedia.org/wiki/Centroid]https://en.wikipedia.org/wiki/Centroid[/url] .[br][color=#333333]The centroid or Geometric Center [/color]of figure is the arithmetic mean position [math]\vec{P_i}[/math] of all n- points in the figure:[br] [math]\vec{C_m}=\frac{\sum\vec{P_i}}{n},i=1,...,n[/math]. Two special expressions are associated with centroid.[br] - From its definition: [math]\sum\left(\vec{C_m}-\vec{P_i}\right)=0[/math] : The Addition of radius vectors of all points relative to the centroid is zero.[br] - Difference of the two sums: over the squared distances for all points from B and from the centroid is equal to the n times squared distance between centroid and B. It follows, that the sum of the squared distances for all points from the centroid [math]C_m[/math] is the smallest. You can compare results with the Steiner's theorem in the case of unit point masses. [url=https://en.wikipedia.org/wiki/Parallel_axis_theorem]https://en.wikipedia.org/wiki/Parallel_axis_theorem[/url] [br] Creation of this applet was inspired by alfinio [url=https://www.geogebra.org/material/show/id/DZbG9HMZ— February 26, 2015 - 11:36 PM]https://www.geogebra.org/material/show/id/DZbG9HMZ— February 26, 2015 - 11:36 PM[/url] to prove and implement it for more general case.[br] Change the number of particles n in a system, the position of points P[math]_i[/math], B. Make sure that the formula is correct and try again.[br]
– Roman Chijner

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4.

Learn more about Squared Distance

  • 1. 2D. Explanation of Invariance of the sum of squares of distances by using the Steiner theorem.
  • 2. 3D. Explanation of Invariance of the sum of squares of distances by using the Steiner theorem.
  • 3. 2D. Explanation of Invariance of the sum of squares of distances by using the Steiner theorem.
  • 4. triangle Gizmo
  • 5. development on regular sqrs
  • 6. developments on regular polys
  • 7. Symmetry numbers in irregular polygons
  • 8. Symmetry number & conditions in right triangles
  • 9. Symmetrical beauty of polygons
  • 10. beauty of powers-2b
  • 11. Sliding trigon
  • 12. Analogy between statics & geometry
  • 13. Analogy between statics & geometry
  • 14. Hidden symmetry in an Atom
  • 15. Ultimate symmetry in 3D
  • 16. symmetry of two points in sph. & cyldr.
  • 17. Ultimate symmetry in cone-sphere sys.
  • 18. Symmetry in sphere & dodecahedron case
  • 19. 1-Concentric icosahedron & sphere
  • 20. Concentric dodecahedron & sphere
  • 21. 3D correspondence of symmetry spheres-B
  • 22. Symmetry number for 10 points
  • 23. Symmetry number for 5 points
  • 24. General solution for symmetry numbers on surfaces-B
  • 25. Symmetry number of a complex system
  • 26. Symmetry number of a complex system
  • 27. symmetry number for two overlapped ellipses
  • 28. David star overlapped
  • 29. symmetry numbers & parabolic mirror
  • 30. Mobilis in mobili symmetry
  • 31. Mobilis in mobili symmetry (generalised)
  • 32. Plane reg. polygon in a sphere
  • 33. Symmetry in random movables
  • 34. Beautiful regular pentagon-B
  • 35. Beautiful regular pentagon-A
  • 36. 4 - Tetrahedron
  • 37. 3 - Octahedron
  • 38. 2 - Icosahedron
  • 39. 1 - Dodecahedron
  • 40. 0 - Definition of platonic sets
  • 41. Plane equilateral triang. & sphere
  • 42. Regular octagon & sphere
  • 43. Symmetry number for double octagons-A
  • 44. Symmetry number for double octagons-B
  • 45. Three squares in a screen
  • 46. Three squares in a screen overlapped
  • 47. Definition of rules for symmetry
  • 48. Definition of rules for symmetry-B
  • 49. Password generator-A
  • 50. Password generator-B
  • 51. Extremely high powers
  • 52. Exremely high powers
  • 53. Extremely high powers
  • 54. Comments on extremely high powers
  • 55. Difference of extremely high powers-5
  • 56. An objection to transcendental definition
  • 57. New theorem of Pythagoras
  • 58. new theorem of pythagoras for circumferences
  • 59. new theorem of pythagoras for non-similar trigons
  • 60. New theorem of pythagoras for non-similar circles
  • 61. Game of delicate balance
  • 62. Puzzle-4
  • 63. Puzzle-2
  • 64. Ponzi game
  • 65. Symmetry of peripheral circles
  • 66. diplomatic relations
  • 67. Cable-car functions
  • 68. Four birds on a line
  • 69. Proxima Centauri
  • 70. Hexagons around hexagon
  • 71. Canis Major
  • 72. Canis Major-A
  • 73. Sagittarius-3D
  • 74. Sagittarius 3D generalized
  • 75. Andromeda
  • 76. imami azam EBU HANIFE
  • 77. Piscis Austrinus
  • 78. Pegasus
  • 79. Cygnus
  • 80. Suma de distancias al cuadrado constante
  • 81. Suma de cuadrados constante
  • 82. Quadratsumme der Radialstrecken
  • 83. Summe der 4.Potenz der Diagonalen
  • 84. Fünfeck mit Kreis (Baryzentrum)

4.1.

2D. Explanation of Invariance of the sum of squares of distances by using the Steiner theorem.

n/2 points of [color=#0000ff]A[sub]1[/sub],...,A[sub]n[/sub][/color] may be moving in a circle of radius r. Each [color=#0000ff]point[/color] has [color=#ff7700]antipodal point[/color] on the circle. Thus, the center of gravity [color=#ff00ff]Cm[/color] is the centre of the circle. [br] It is shown, that the sum of squares of distances from any point [color=#ff0000]D[/color] of a circle to → {[color=#0000ff]A[sub]i[/sub][/color] } , divided by radius squared, is always equal of twice the number of points. [br][url=https://en.wikipedia.org/wiki/Parallel_axis_theorem]https://en.wikipedia.org/wiki/Parallel_axis_theorem[/url][br][url=https://en.wikipedia.org/wiki/Antipodal_point]https://en.wikipedia.org/wiki/Antipodal_point[/url]
– Roman Chijner

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5.

Symmedians

  • 1. X(85) Isogonal conjugate of X(41)
  • 2. Isogonal Conjugates Proof of Existence (Symmedians)
  • 3. Isogonal Conjugate of line through Centroid
  • 4. The Kiepert Hyperbola and the Brocard Axis
  • 5. Symmedian Point

5.1.

X(85) Isogonal conjugate of X(41)

isogonal conjugate of X(41)
Triangle center X(41) is the X(6)-ceva conjugate of X(31).[br]X(6) is the symmedian point[br]X(31) is the 2nd power point of the triangle ABC.[br]A power point is a point whose coordinates are defined by a power function.[br]For the 2nd power point the [url=http://mathworld.wolfram.com/TrilinearCoordinates.html]trilinear coordinates[/url] are a² : b² : c².[br]P, triangle center X(41) is the X(6)-[url=http://mathworld.wolfram.com/CevaConjugate.html]Ceva conjugate[/url] of X(31). This means that P, the X(6)-Ceva conjugate of X(31) is given by the perspector of the Cevian triangle of X(6) and the anticevian triangle of X(31).[br]The isogonal conjugate of X[sub]41[/sub], triangle center X(41) can be constructed as follows:[br][list][*]Reflect the lines AX[sub]41[/sub], BX[sub]41[/sub], CX[sub]41[/sub] about the bisectors of the triangle ABC (=blue lines)[/*][*]These blue lines cross at the triangle center X(85).[/*][/list]The barycentric coordinates of this point depend on the lenghts of the triangle.
isogonale toegevoegde van X(41)
Driehoekscentrum X(41) is de X(6)-ceva toegevoegde van X(31).[br]X(6) is het punt van Lemoine.[br]X(31) is het tweedemachtspunt van de driehoek ABC.[br]Een machtspunt is een punt waarvan de coördintaten bepaald worden door een machtsfunctie.[br]Voor het tweedemachtspunt worden de [url=http://mathworld.wolfram.com/TrilinearCoordinates.html]trilineaire coördinaten[/url] a² : b² : c².[br]P, driehoekscentrum X(41) is de X(6) [url=http://mathworld.wolfram.com/CevaConjugate.html]Ceva toegevoegde[/url] van X(31). Dit betekent dat P, de X(6)-Ceva toegevoegde X(31) het perspectiefcentrum is va de Ceva driehoek van X(6) en de anticeva driehoek van X(31).[br]Het isogonale toegevoegde punt van X[sub]41[/sub], het driehoekscentrum X(41) construeer je als volgt:[br][list][*]Spiegel de rechten AX[sub]41[/sub], BX[sub]41[/sub], CX[sub]41[/sub] t.o.v. de bissectrices van ABC (=blauwe lijnen).[/*][*]Deze blauwe lijnen snijden elkaar in het driehoekscentrum X(85).[/*][/list]De barycentrische coördinaten van dit punt worden bepaald door de lengtes van de zijden van de driehoek.
– chris cambré

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