Illustrations Without Words

Tim Brzezinski, Wellerson Davi, 08/lug/2018

Contains many dynamic illustrations WITHOUT words, segment lengths, and angle measures. In these applets, the slider governs all applet dynamics. These applets can be used to help a student informally & visually discover a mathematics concept and/or meaningfully recall a certain concept. Enjoy!

Sommario

  1. Writing Prompts: Definitions & Theorems
    1. Congruent Segments: Quick Exploration
    2. Congruent Angles: Definition
    3. Midpoint Definition
    4. Segment Bisector Definition
    5. Perpendicular Bisector Definition
    6. Perpendicular Bisector Definition
    7. Angle Bisector Definition
    8. Complementary Angles Meaning
    9. Supplementary Angles Meaning
    10. Triangle Altitude Illustrator and Definition Writing Prompt
    11. Triangle Median Illustrator & Definition Writing Prompt
    12. Equilateral Action!!!
    13. Equiangular Action!!!
    14. Regular Polygon Action!
    15. Proof Prompt (A)
    16. Proof Prompt (B)
    17. Proof Prompt (C)
    18. Proof Prompt (D)
    19. A Special Theorem: Part 2 (V2)
    20. A Special Theorem: Part 2 (V3)
    21. A Special Theorem: Part 2 (V4)
    22. Another Special Theorem: Part 1 (V2)
    23. Another Special Theorem: Part 1 (V3)
    24. Slope Triangle Rotation (VA)
    25. Slope Triangle Rotation (VB)
  2. Parallel Lines & Related Angles
    1. Vertical Angles Theorem
    2. Corresponding Angles and Parallel Lines
    3. Alternate Interior Angles and Parallel Lines
    4. Exploring Alternate Interior Angles (V2)
    5. Alternate Interior Angles Theorem (V3)
    6. Alternate Exterior Angles and Parallel Lines
    7. Exploring Alternate Exterior Angles (V2)
    8. Alternate Exterior Angles Theorem (V3)
    9. Same Side Interior Angles and Parallel Lines
    10. Same Side Exterior Angles and Parallel Lines
    11. Solution to Parallel Lines & Angles Problem
    12. Effortless Trisections
    13. Special Pair of Lines in the Coordinate Plane
  3. Polygons & Angles: Interior & Exterior Angles
    1. Triangle Angle Theorems
    2. Triangle Angle Theorems (V2)
    3. Triangle Angle Theorems (V3)
    4. Obvious Corollary
    5. Triangle Exterior Angle
    6. Angles of a Right Triangle
    7. Acute Angles of a Right Triangle
    8. Quadrilateral Angle Theorems
    9. Pentagon Angle Theorems
    10. Hexagon Angle Theorems
    11. Heptagon Angle Theorems
    12. Octagon Angle Theorems
    13. Exploring Polygon Angles: Triangle through Octagon
    14. GoGeometry Action 40!
    15. GoGeometry Action 58!
    16. SolveMyMaths Action 2!
  4. Polygons: Exterior Angles Only
    1. Triangle: Exterior Angles
    2. Exterior Angles of a Triangle
    3. Quadrilateral: Exterior Angles
    4. Exterior Angles of a Quadrilateral
    5. Pentagon: Exterior Angles
    6. Polygons: Exterior Angles
    7. Polygons: Exterior Angles - REVAMPED
    8. Exterior Angles (Revisited)
    9. Exterior Angles of a Pentagon
    10. Exterior Angles of a Hexagon
    11. Exterior Angles of a Heptagon
    12. Exterior Angles of an Octagon
    13. Exterior Angles (Revisited) V2
  5. Quadrilaterals: Theorems & Properties
    1. Parallelogram: Theorem 1
    2. Parallelogram: Theorem 2
    3. Parallelogram: Theorem (3)
    4. Parallelograms (I)
    5. Parallelogram Diagonals
    6. Special Quadrilateral: Theorem 1
    7. Special Quadrilateral: Theorem 2
    8. Special Quadrilateral: Theorem 3
    9. Special Quadrilateral: Theorem 4
    10. Quad Midpoints Action!
    11. Rhombus Action!
    12. Rhombus Diagonals
    13. Rectangle Action!
    14. Rectangle Diagonals
    15. Square Action!
    16. Isosceles Trapezoid Action!
    17. Trapezoid Median: 2 Discoveries
    18. Isosceles Trapezoid Diagonals
    19. Isosceles Trapezoid Midpoints
    20. Kite Action!
    21. SSSS: Similar Quads?
    22. AAA: Similar Quads?
    23. Trapezoid Median (Midsegment) Action!
    24. Thebault Action!!!
    25. Square Problem
    26. 3 Squares Problem
    27. Square Action! (Proof Problem 1)
    28. Harmonic Mean In an Isosceles Trapezoid
    29. Cut-The-Knot Action (4)!
    30. Cut-The-Knot-Action (5)!
    31. Rhombus Action + Sequel = GoGeometry Action 7!
    32. GoGeometry Action 11!
    33. GoGeometry Action 12!
    34. GoGeometry Action 18!
    35. GoGeometry Action 21!
    36. GoGeometry Action 22!
    37. GoGeometry Action 23!
    38. GoGeometry Action 24!
    39. GoGeometry Action 36!
    40. Cut-the-Knot-Action 13!
    41. GoGeometry Action 42!
    42. Cut-The-Knot-Action 14!
    43. GoGeometry Action 46!
    44. GoGeometry Action 47!
    45. Rhombus Surprise! (GoGeometry Action 57)
    46. GoGeometry Action 67!
    47. GoGeometry Action 69!
    48. Square + Rhombus = Constant Surprise! (GoGeometry Action 71)
    49. GoGeometry Action 73!
    50. Parallelogram? Why So? (GoGeometry Action 77)
    51. Concave Quadrilateral Craziness! (GoGeometry Action 80)
    52. Square and Quarter Circle Action = GoGeometry Action 81!
    53. GoGeometry Action 94!
    54. GoGeometry Action 95!
    55. GoGeometry Action 96!
    56. GoGeometry Action 98!
    57. GoGeometry Action 99!
    58. GoGeometry Action 102 !
    59. GoGeometry Action 108!
    60. GoGeometry Action 114!
    61. GoGeometry Action 119!
    62. GoGeometry Action 126!
    63. GoGeometry Action 130!
    64. GoGeometry Action 135!
    65. GoGeometry Action 136!
    66. GoGeometry Action 139!
    67. GoGeometry Action 140!
    68. Parallelogram + 2 Equilateral Triangles: How to Prove?
    69. GoGeometry Action 143!
    70. GoGeometry Action 153!
  6. Triangles: Theorems & Properties
    1. SAS: Dynamic Proof!
    2. SSS: Dynamic Proof!
    3. SSS: Dynamically Illustrated
    4. ASA Theorem?
    5. HL: Hypotenuse-Leg Action!
    6. Similar Right Triangles (V1)
    7. Similar Right Triangles (V2)
    8. AA Similarity Theorem
    9. SAS ~ Theorem
    10. SSS ~ Theorem (V1)
    11. SSS ~ Theorem (V2)
    12. Parallel Lines Proportionality Theorem
    13. Triangle-Angle Bisector Theorem
    14. A Special Triangle & Its Properties (I)
    15. Converse of IST (V1)
    16. Isosceles Triangle Medians
    17. Special Line through Triangle V1 (Theorem Discovery)
    18. Special Line through Triangle V2 (Theorem Discovery)
    19. Triangle Midsegment Action!
    20. Triangle Midsegment
    21. Another Special Triangle and its Properties (II)
    22. Converse of IST (II)
    23. Isosceles Triangle: Surprising Theorem!
    24. Special Right Triangle (I)
    25. Special Right Triangle (II)
    26. Another Special Right Triangle (Guided Discovery)
    27. Incenter & Incircle Action!
    28. Incenter + Incircle Action (V2)!
    29. Medians and Centroid Dance
    30. Circumcenter & Circumcircle Action!
    31. Midpoint of HYP
    32. 3 Special Points!
    33. 9 Point Circle Action
    34. 9-Point Circle Action (Part 2)
    35. 9-Point Circle Action (Part 3B)
    36. 9-Point Circle Action (Part 4)
    37. 9-Point Circle Action (Part 5)
    38. More Triangle Action!!!
    39. Triangle Median & Circles Action!!!
    40. 4 Triangles Action!!!
    41. Desargues Action!!!
    42. GoGeometry Action 10!
    43. GoGeometry Action 14!
    44. GoGeometry Action 17!
    45. GoGeometry Action 18!
    46. Cut-The-Knot-Action 8!
    47. GoGeometry Action 20!
    48. Cut-The-Knot Action 9!
    49. Carnot Action! = GoGeometry Action 27!
    50. GoGeometry Action 29!
    51. GoGeometry Action 30!
    52. Incircle + Excircle = GoGeometry Action 31!
    53. GoGeometry Action 32!
    54. Cut-The-Knot Action 10!
    55. Finsler-Hadwiger Sequel! (GoGeometry Action 33)
    56. GoGeometry Action 34!
    57. Cut-The-Knot-Action 12!
    58. GoGeometry Action 35!
    59. GoGeometry Action 37!
    60. GoGeometry Action 38!
    61. GoGeometry Action 39!
    62. GoGeometry Action 41!
    63. GoGeometry Action 43!
    64. GoGeometry Action 44!
    65. GoGeometry Action 45!
    66. GoGeometry Action 51!
    67. GoGeometry Action 53!
    68. Euler Line Surprise! = GoGeometry Action 54
    69. GoGeometry Action 60!
    70. GoGeometry Action 63!
    71. GoGeometry Action 64!
    72. GoGeometry Action 66!
    73. GoGeometry Action 68!
    74. GoGeometry Action 74!
    75. GoGeometry Action 75!
    76. GoGeometry Action 76!
    77. GoGeometry Action 78!
    78. GoGeometry Action 79!
    79. GoGeometry Action 82!
    80. GoGeometry Action 83!
    81. Cut-the-Knot Action 15!
    82. Equilateral Triangle + Semicircle = ? (GoGeometry Action 88)
    83. GoGeometry Action 89!
    84. GoGeometry Action 90!
    85. GoGeometry Action 91!
    86. GoGeometry Action 93!
    87. GoGeometry Action 97!
    88. GoGeometry Action 101 !
    89. GoGeometry Action 103!
    90. GoGeometry Action 104!
    91. GoGeometry Action 105!
    92. GoGeometry Action 106!
    93. GoGeometry Action 107!
    94. GoGeometry Action 109!
    95. SolveMyMaths Action 1!
    96. Viviani Action! (REVAMPED)
    97. GoGeometry Action 110!
    98. GoGeometry Action 112!
    99. GoGeometry Action 113!
    100. GoGeometry Action 115!
    101. GoGeometry Action 116!
    102. GoGeometry Action 117!
    103. Blanchet Action = GoGeometry Action 118!
    104. GoGeometry Action 120!
    105. GoGeometry Action 121!
    106. GoGeometry Action 124!
    107. GoGeometry Action 125!
    108. Cut-The-Knot Action 18!
    109. Cut-The-Knot Action 19!
    110. GoGeometry Action 127!
    111. Incircle Extension = GoGeometry Action 132!
    112. Incircle Extension 2 = GoGeometry Action 133!
    113. GoGeometry Action 134!
    114. GoGeometry Action 138!
    115. GoGeometry Action 147!
    116. GoGeometry Action 152!
    117. GoGeometry Action 156!
    118. GoGeometry Action 160!
    119. GoGeometry Action 161!
    120. GoGeometry Action 162!
  7. Circle Theorems
    1. Congruent Circles: Definition
    2. Circle Tangent-Types Illustrator
    3. Properties of Tangents Drawn to Circles (A)
    4. Properties of Tangents Drawn to Circles (B)
    5. Int+Ext Tangent Action!
    6. Diameters & Chords I (VA)
    7. Diameters & Chords I (VB)
    8. Congruent Chords: Quick Investigation
    9. Congruent Chords (I)
    10. Inscribed Angle Theorem (V1)
    11. Find that Angle! (GoGeometry Action 144)
    12. Proof Exercise: Inscribed Angle Theorem (Case 1)
    13. Proof Exercise: Inscribed Angle Theorem (Case 2)
    14. Proof Exercise: Inscribed Angle Theorem (Case 3)
    15. Inscribed Angle Theorem Dance: Take 2!
    16. Inscribed Angle Theorem: Corollary 1
    17. Where to Sit? (I)
    18. Thales' Theorem (VA)
    19. Thales' Theorem (VB)
    20. Inscribed Angle Intercepts Semicircle
    21. Cyclic Quadrilaterals (IAT: Corollary 3)
    22. Cyclic Quadrilateral: Proof Hint
    23. Cyclic Quadrilateral: Quick Warm Up (GoGeometry Action 149!)
    24. Angle Formed by Chord & Tangent (I)
    25. Angle Formed by Chord & Tangent (II)
    26. Angle Formed by 2 Chords (I)
    27. Angle Formed by 2 Chords (II)
    28. Crossing Chords: Proof Hint
    29. Angles from Secants and Tangents (V1)
    30. Angle From 2 Secants (V2)
    31. Secants: Proof Hint
    32. Congruent Chords Action (A)
    33. Equidistant Chord Action
    34. Butterfly Theorem Action!
    35. Not Your Everyday Chord & Tangent Theorem
    36. Cut-The-Knot Action (2) !!!
    37. Square, Tangent, and Two Congruent & Tangent Circles
    38. GoGeometry Action 1!
    39. GoGeometry Action 2!
    40. GoGeometry Action 3!
    41. GoGeometry Action 4!
    42. GoGeometry Action 5!
    43. Geometric Mean Illustration
    44. GoGeometry Action 6!
    45. GoGeometry Action 8!
    46. Cut-The-Knot Action (6)!
    47. GoGeometry Action 9!
    48. GoGeometry Action 13!
    49. GoGeometry Action 15!
    50. GoGeometry Action 16!
    51. Cut-The-Knot Action 7!
    52. GoGeometry Action 19!
    53. Thales Action + Sequel = GoGeometry Action 25!
    54. GoGeometry Action 26!
    55. GoGeometry Action 48!
    56. GoGeometry Action 49!
    57. GoGeometry Action 50!
    58. GoGeometry Action 52!
    59. 60-Degree Sector Surprise! (GoGeometry Action 55)
    60. Neusis Logic Action = GoGeometry Action 56!
    61. Quarter Circle Surprise!
    62. GoGeometry Action 59!
    63. GoGeometry Action 61!
    64. GoGeometry Action 62!
    65. GoGeometry Action 70!
    66. GoGeometry Action 72!
    67. Geometric Mean Madness! (GoGeometry Action 84)
    68. Mickey Mouse Theorem! (Cut-the-Knot Action 16)
    69. More Geometric Mean Action! (GoGeometry Action 86)
    70. GoGeometry Action 87!
    71. GoGeometry Action 92!
    72. A Stronger Conclusion!
    73. GoGeometry Action 100!
    74. GoGeometry Action 122!
    75. GoGeometry Action 128!
    76. GoGeometry Action 129!
    77. GoGeometry Action 131!
    78. GoGeometry Action 137!
    79. GoGeometry Action 141!
    80. GoGeometry Action 142!
    81. GoGeometry Action 148!
    82. Archimedes Insight = GoGeometry Action 150!
    83. More Archimedes Insight = GoGeometry Action 151!
    84. GoGeometry Action 154!
    85. GoGeometry Action 155!
    86. Archimedes Insight 3: GoGeometry Action 157!
  8. Geometric Constructions
    1. Isosceles Triangle (Quick Construction Technique) V1
    2. Equilateral Triangle Construction (Dynamic Illustration)
    3. Congruent Angle Construction Action!
    4. Angle Bisector Construction Action!
    5. Alternate Angle Bisector Construction
    6. Bizzare Trisection?
    7. Angle Trisection by Paper Folding!
    8. Bizarre Trisection (V2) = GoGeometry Action 145!
    9. Bizarre Trisection (V3) = GoGeometry Action 146!
  9. Transformations
    1. Dilating a Point (Intro)
    2. Dilating a Line: HSG.SRT.A.1.A
    3. Dilating a Segment: HSG.SRT.A.1.B
    4. Properties of Dilations
  10. Area, Surface Area, Volume
    1. Parallelogram: Area
    2. Triangle Area Action! (V1)
    3. Area of a Triangle (Discovery)
    4. Trapezoid: Area (I)
    5. Area of a Trapezoid (Discovery)
    6. Scaling Figures: Effects on Areas
    7. SolveMyMaths Action 3A!
    8. SolveMyMaths Action 3B!
    9. SolveMyMaths Action 4!
    10. Right Circular Cylinder!
  11. Slope
    1. Parallel Consequence: Quick Reminder
    2. Parallel Consequence? (VA)
    3. Parallel Consequence? (VB)
    4. Parallel Lines in the Coordinate Plane (VA)
    5. Parallel Lines in the Coordinate Plane (VB)
    6. Perpendicular Lines in the CP: Quick Investigation
    7. Slope Triangle Rotation (VA)
    8. Slope Triangle Rotation (VB)
  12. Functions
    1. Functions: Input & Output (3)
    2. Functions: Input & Output (4)
    3. Half-Life Action!!!
    4. What Makes a Function Even?
    5. What Makes a Function Odd?
    6. Odd Functions (Another Look)
    7. Odd Functions: Another Look (Take 2)!
    8. Function Composition: Dynamic Illustrator (1)
    9. Periodic Function Action!
    10. Logarithmic Action!
    11. Logarithmic Action (2)!
    12. Logarithmic Action (3)!
    13. Logarithmic Action (4)!
    14. Logarithmic Action (4)! V2
  13. Vectors
    1. Vector Addition
    2. Vector Subtraction
    3. Scaling Vectors
    4. Vector Projections (1)
  14. Conic Sections: Definitions & Properties
    1. Parabola as a Locus
    2. Parabola: Cause and Effect
    3. Special Conic LR Action
    4. Parabola: Geometric Property (I)
    5. Parabola: Geometric Property (II)
    6. Another Parabola Theorem!
    7. Ellipse: Magic Sum = ?
    8. Ellipse: Magic Sum = ? (V2)
    9. Ellipse: Reflective Property
    10. Ellipse: Reflective Property
    11. Hyperbola: Difference = ?
    12. Hyperbola: Reflective Property
  15. Trigonometry
    1. Angle Orientations (Standard Position)
    2. Coterminal Angles Action!!!
    3. Special Angles in Standard Position: Intro (V1)
    4. Special Angles in Standard Position: Intro (V2)
    5. Special Angles in Standard Position: Intro (V2) - Part 2
    6. Special Angles in Standard Position: Intro (V3)
    7. Sine & Cosine Period Action (2)!
    8. Tangent Period Action (2)!
    9. Right Triangle Trigonometry: Intro
    10. Trigonometric Ratios (Right Triangle Context)
    11. What "CO" in COsine Means
    12. Law of Sines (& Area)
    13. Law of Cosines: Discovery
    14. Bizzare Trig Identity?
    15. 3 Squares Problem
    16. Polar Coordinates: Introduction
  16. Illustrations in 3-Space
    1. Non-Coplanar "Quad" Midpoints?
    2. Sphere Investigation
    3. Circular Paraboloid of Revolution
    4. Special Spheroid Action: Locus Illustration
    5. 3 Golden Rectangles Surprise!
    6. Dodecahedral Surprise!
    7. Dandelin Sphere Action!
  17. Calculus Applets
    1. Derivative Function: Without Words
    2. Area Between Curves
    3. Mean Value Theorem (Without Words)
  18. Cool But Not-Often-Seen Theorems
    1. Equilateral Action (1)!
    2. Equilateral Action (2)!
    3. Equilateral Action (3)!
    4. Napoleon Motion!
    5. Hexagonal Napoleon Theorem?
    6. Triangle to Conic?
    7. Square Problem
    8. Japanese Theorem for Cyclic Quadrilaterals
    9. Cyclic Quad Action (II)
    10. Equilateral Action (4)!
    11. Equilateral Triangle and Rectangle (Area Problem)
    12. Van Aubel Action!!!
    13. Finsler-Hadwiger Action!!!
    14. Morley Action!
    15. Quadrilateral Surprise!
    16. Cut-The-Knot Action !!!
    17. Cut-the-Knot-Action (3)!
    18. Cut-The-Knot-Action 11!
    19. GoGeometry Action 65!
    20. Hexagonal Surprise! (GoGeometry Action 85)
    21. GoGeometry Action 111!
    22. GoGeometry Action 123!
    23. Cut-The-Knot Action 17!
    24. From Triangle to WHAT?
    25. From Equiangular Hexagon to WHAT?
    26. GoGeometry Action 158!
    27. GoGeometry Action 159!
    28. Regular Decagon Magic!

1.

Writing Prompts: Definitions & Theorems

  • 1. Congruent Segments: Quick Exploration
  • 2. Congruent Angles: Definition
  • 3. Midpoint Definition
  • 4. Segment Bisector Definition
  • 5. Perpendicular Bisector Definition
  • 6. Perpendicular Bisector Definition
  • 7. Angle Bisector Definition
  • 8. Complementary Angles Meaning
  • 9. Supplementary Angles Meaning
  • 10. Triangle Altitude Illustrator and Definition Writing Prompt
  • 11. Triangle Median Illustrator & Definition Writing Prompt
  • 12. Equilateral Action!!!
  • 13. Equiangular Action!!!
  • 14. Regular Polygon Action!
  • 15. Proof Prompt (A)
  • 16. Proof Prompt (B)
  • 17. Proof Prompt (C)
  • 18. Proof Prompt (D)
  • 19. A Special Theorem: Part 2 (V2)
  • 20. A Special Theorem: Part 2 (V3)
  • 21. A Special Theorem: Part 2 (V4)
  • 22. Another Special Theorem: Part 1 (V2)
  • 23. Another Special Theorem: Part 1 (V3)
  • 24. Slope Triangle Rotation (VA)
  • 25. Slope Triangle Rotation (VB)

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2.

Parallel Lines & Related Angles

  • 1. Vertical Angles Theorem
  • 2. Corresponding Angles and Parallel Lines
  • 3. Alternate Interior Angles and Parallel Lines
  • 4. Exploring Alternate Interior Angles (V2)
  • 5. Alternate Interior Angles Theorem (V3)
  • 6. Alternate Exterior Angles and Parallel Lines
  • 7. Exploring Alternate Exterior Angles (V2)
  • 8. Alternate Exterior Angles Theorem (V3)
  • 9. Same Side Interior Angles and Parallel Lines
  • 10. Same Side Exterior Angles and Parallel Lines
  • 11. Solution to Parallel Lines & Angles Problem
  • 12. Effortless Trisections
  • 13. Special Pair of Lines in the Coordinate Plane

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3.

Polygons & Angles: Interior & Exterior Angles

  • 1. Triangle Angle Theorems
  • 2. Triangle Angle Theorems (V2)
  • 3. Triangle Angle Theorems (V3)
  • 4. Obvious Corollary
  • 5. Triangle Exterior Angle
  • 6. Angles of a Right Triangle
  • 7. Acute Angles of a Right Triangle
  • 8. Quadrilateral Angle Theorems
  • 9. Pentagon Angle Theorems
  • 10. Hexagon Angle Theorems
  • 11. Heptagon Angle Theorems
  • 12. Octagon Angle Theorems
  • 13. Exploring Polygon Angles: Triangle through Octagon
  • 14. GoGeometry Action 40!
  • 15. GoGeometry Action 58!
  • 16. SolveMyMaths Action 2!

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4.

Polygons: Exterior Angles Only

  • 1. Triangle: Exterior Angles
  • 2. Exterior Angles of a Triangle
  • 3. Quadrilateral: Exterior Angles
  • 4. Exterior Angles of a Quadrilateral
  • 5. Pentagon: Exterior Angles
  • 6. Polygons: Exterior Angles
  • 7. Polygons: Exterior Angles - REVAMPED
  • 8. Exterior Angles (Revisited)
  • 9. Exterior Angles of a Pentagon
  • 10. Exterior Angles of a Hexagon
  • 11. Exterior Angles of a Heptagon
  • 12. Exterior Angles of an Octagon
  • 13. Exterior Angles (Revisited) V2

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5.

Quadrilaterals: Theorems & Properties

  • 1. Parallelogram: Theorem 1
  • 2. Parallelogram: Theorem 2
  • 3. Parallelogram: Theorem (3)
  • 4. Parallelograms (I)
  • 5. Parallelogram Diagonals
  • 6. Special Quadrilateral: Theorem 1
  • 7. Special Quadrilateral: Theorem 2
  • 8. Special Quadrilateral: Theorem 3
  • 9. Special Quadrilateral: Theorem 4
  • 10. Quad Midpoints Action!
  • 11. Rhombus Action!
  • 12. Rhombus Diagonals
  • 13. Rectangle Action!
  • 14. Rectangle Diagonals
  • 15. Square Action!
  • 16. Isosceles Trapezoid Action!
  • 17. Trapezoid Median: 2 Discoveries
  • 18. Isosceles Trapezoid Diagonals
  • 19. Isosceles Trapezoid Midpoints
  • 20. Kite Action!
  • 21. SSSS: Similar Quads?
  • 22. AAA: Similar Quads?
  • 23. Trapezoid Median (Midsegment) Action!
  • 24. Thebault Action!!!
  • 25. Square Problem
  • 26. 3 Squares Problem
  • 27. Square Action! (Proof Problem 1)
  • 28. Harmonic Mean In an Isosceles Trapezoid
  • 29. Cut-The-Knot Action (4)!
  • 30. Cut-The-Knot-Action (5)!
  • 31. Rhombus Action + Sequel = GoGeometry Action 7!
  • 32. GoGeometry Action 11!
  • 33. GoGeometry Action 12!
  • 34. GoGeometry Action 18!
  • 35. GoGeometry Action 21!
  • 36. GoGeometry Action 22!
  • 37. GoGeometry Action 23!
  • 38. GoGeometry Action 24!
  • 39. GoGeometry Action 36!
  • 40. Cut-the-Knot-Action 13!
  • 41. GoGeometry Action 42!
  • 42. Cut-The-Knot-Action 14!
  • 43. GoGeometry Action 46!
  • 44. GoGeometry Action 47!
  • 45. Rhombus Surprise! (GoGeometry Action 57)
  • 46. GoGeometry Action 67!
  • 47. GoGeometry Action 69!
  • 48. Square + Rhombus = Constant Surprise! (GoGeometry Action 71)
  • 49. GoGeometry Action 73!
  • 50. Parallelogram? Why So? (GoGeometry Action 77)
  • 51. Concave Quadrilateral Craziness! (GoGeometry Action 80)
  • 52. Square and Quarter Circle Action = GoGeometry Action 81!
  • 53. GoGeometry Action 94!
  • 54. GoGeometry Action 95!
  • 55. GoGeometry Action 96!
  • 56. GoGeometry Action 98!
  • 57. GoGeometry Action 99!
  • 58. GoGeometry Action 102 !
  • 59. GoGeometry Action 108!
  • 60. GoGeometry Action 114!
  • 61. GoGeometry Action 119!
  • 62. GoGeometry Action 126!
  • 63. GoGeometry Action 130!
  • 64. GoGeometry Action 135!
  • 65. GoGeometry Action 136!
  • 66. GoGeometry Action 139!
  • 67. GoGeometry Action 140!
  • 68. Parallelogram + 2 Equilateral Triangles: How to Prove?
  • 69. GoGeometry Action 143!
  • 70. GoGeometry Action 153!

5.1.

–

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

5.9.

–

5.10.

–

5.11.

–

5.12.

–

5.13.

–

5.14.

–

5.15.

–

5.16.

–

5.17.

–

5.18.

–

5.19.

–

5.20.

–

5.21.

–

5.22.

–

5.23.

–

5.24.

–

5.25.

–

5.26.

–

5.27.

–

5.28.

–

5.29.

–

5.30.

–

5.31.

–

5.32.

–

5.33.

–

5.34.

–

5.35.

–

5.36.

–

5.37.

–

5.38.

–

5.39.

–

5.40.

–

5.41.

–

5.42.

–

5.43.

–

5.44.

–

5.45.

–

5.46.

–

5.47.

–

5.48.

–

5.49.

–

5.50.

–

5.51.

–

5.52.

–

5.53.

–

5.54.

–

5.55.

–

5.56.

–

5.57.

–

5.58.

–

5.59.

–

5.60.

–

5.61.

–

5.62.

–

5.63.

–

5.64.

–

5.65.

–

5.66.

–

5.67.

–

5.68.

–

5.69.

–

5.70.

–

6.

Triangles: Theorems & Properties

  • 1. SAS: Dynamic Proof!
  • 2. SSS: Dynamic Proof!
  • 3. SSS: Dynamically Illustrated
  • 4. ASA Theorem?
  • 5. HL: Hypotenuse-Leg Action!
  • 6. Similar Right Triangles (V1)
  • 7. Similar Right Triangles (V2)
  • 8. AA Similarity Theorem
  • 9. SAS ~ Theorem
  • 10. SSS ~ Theorem (V1)
  • 11. SSS ~ Theorem (V2)
  • 12. Parallel Lines Proportionality Theorem
  • 13. Triangle-Angle Bisector Theorem
  • 14. A Special Triangle & Its Properties (I)
  • 15. Converse of IST (V1)
  • 16. Isosceles Triangle Medians
  • 17. Special Line through Triangle V1 (Theorem Discovery)
  • 18. Special Line through Triangle V2 (Theorem Discovery)
  • 19. Triangle Midsegment Action!
  • 20. Triangle Midsegment
  • 21. Another Special Triangle and its Properties (II)
  • 22. Converse of IST (II)
  • 23. Isosceles Triangle: Surprising Theorem!
  • 24. Special Right Triangle (I)
  • 25. Special Right Triangle (II)
  • 26. Another Special Right Triangle (Guided Discovery)
  • 27. Incenter & Incircle Action!
  • 28. Incenter + Incircle Action (V2)!
  • 29. Medians and Centroid Dance
  • 30. Circumcenter & Circumcircle Action!
  • 31. Midpoint of HYP
  • 32. 3 Special Points!
  • 33. 9 Point Circle Action
  • 34. 9-Point Circle Action (Part 2)
  • 35. 9-Point Circle Action (Part 3B)
  • 36. 9-Point Circle Action (Part 4)
  • 37. 9-Point Circle Action (Part 5)
  • 38. More Triangle Action!!!
  • 39. Triangle Median & Circles Action!!!
  • 40. 4 Triangles Action!!!
  • 41. Desargues Action!!!
  • 42. GoGeometry Action 10!
  • 43. GoGeometry Action 14!
  • 44. GoGeometry Action 17!
  • 45. GoGeometry Action 18!
  • 46. Cut-The-Knot-Action 8!
  • 47. GoGeometry Action 20!
  • 48. Cut-The-Knot Action 9!
  • 49. Carnot Action! = GoGeometry Action 27!
  • 50. GoGeometry Action 29!
  • 51. GoGeometry Action 30!
  • 52. Incircle + Excircle = GoGeometry Action 31!
  • 53. GoGeometry Action 32!
  • 54. Cut-The-Knot Action 10!
  • 55. Finsler-Hadwiger Sequel! (GoGeometry Action 33)
  • 56. GoGeometry Action 34!
  • 57. Cut-The-Knot-Action 12!
  • 58. GoGeometry Action 35!
  • 59. GoGeometry Action 37!
  • 60. GoGeometry Action 38!
  • 61. GoGeometry Action 39!
  • 62. GoGeometry Action 41!
  • 63. GoGeometry Action 43!
  • 64. GoGeometry Action 44!
  • 65. GoGeometry Action 45!
  • 66. GoGeometry Action 51!
  • 67. GoGeometry Action 53!
  • 68. Euler Line Surprise! = GoGeometry Action 54
  • 69. GoGeometry Action 60!
  • 70. GoGeometry Action 63!
  • 71. GoGeometry Action 64!
  • 72. GoGeometry Action 66!
  • 73. GoGeometry Action 68!
  • 74. GoGeometry Action 74!
  • 75. GoGeometry Action 75!
  • 76. GoGeometry Action 76!
  • 77. GoGeometry Action 78!
  • 78. GoGeometry Action 79!
  • 79. GoGeometry Action 82!
  • 80. GoGeometry Action 83!
  • 81. Cut-the-Knot Action 15!
  • 82. Equilateral Triangle + Semicircle = ? (GoGeometry Action 88)
  • 83. GoGeometry Action 89!
  • 84. GoGeometry Action 90!
  • 85. GoGeometry Action 91!
  • 86. GoGeometry Action 93!
  • 87. GoGeometry Action 97!
  • 88. GoGeometry Action 101 !
  • 89. GoGeometry Action 103!
  • 90. GoGeometry Action 104!
  • 91. GoGeometry Action 105!
  • 92. GoGeometry Action 106!
  • 93. GoGeometry Action 107!
  • 94. GoGeometry Action 109!
  • 95. SolveMyMaths Action 1!
  • 96. Viviani Action! (REVAMPED)
  • 97. GoGeometry Action 110!
  • 98. GoGeometry Action 112!
  • 99. GoGeometry Action 113!
  • 100. GoGeometry Action 115!
  • 101. GoGeometry Action 116!
  • 102. GoGeometry Action 117!
  • 103. Blanchet Action = GoGeometry Action 118!
  • 104. GoGeometry Action 120!
  • 105. GoGeometry Action 121!
  • 106. GoGeometry Action 124!
  • 107. GoGeometry Action 125!
  • 108. Cut-The-Knot Action 18!
  • 109. Cut-The-Knot Action 19!
  • 110. GoGeometry Action 127!
  • 111. Incircle Extension = GoGeometry Action 132!
  • 112. Incircle Extension 2 = GoGeometry Action 133!
  • 113. GoGeometry Action 134!
  • 114. GoGeometry Action 138!
  • 115. GoGeometry Action 147!
  • 116. GoGeometry Action 152!
  • 117. GoGeometry Action 156!
  • 118. GoGeometry Action 160!
  • 119. GoGeometry Action 161!
  • 120. GoGeometry Action 162!

6.1.

–

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

6.7.

–

6.8.

–

6.9.

–

6.10.

–

6.11.

–

6.12.

–

6.13.

–

6.14.

–

6.15.

–

6.16.

–

6.17.

–

6.18.

–

6.19.

–

6.20.

–

6.21.

–

6.22.

–

6.23.

–

6.24.

–

6.25.

–

6.26.

–

6.27.

–

6.28.

–

6.29.

–

6.30.

–

6.31.

–

6.32.

–

6.33.

–

6.34.

–

6.35.

–

6.36.

–

6.37.

–

6.38.

–

6.39.

–

6.40.

–

6.41.

–

6.42.

–

6.43.

–

6.44.

–

6.45.

–

6.46.

–

6.47.

–

6.48.

–

6.49.

–

6.50.

–

6.51.

–

6.52.

–

6.53.

–

6.54.

–

6.55.

–

6.56.

–

6.57.

–

6.58.

–

6.59.

–

6.60.

–

6.61.

–

6.62.

–

6.63.

–

6.64.

–

6.65.

–

6.66.

–

6.67.

–

6.68.

–

6.69.

–

6.70.

–

6.71.

–

6.72.

–

6.73.

–

6.74.

–

6.75.

–

6.76.

–

6.77.

–

6.78.

–

6.79.

–

6.80.

–

6.81.

–

6.82.

–

6.83.

–

6.84.

–

6.85.

–

6.86.

–

6.87.

–

6.88.

–

6.89.

–

6.90.

–

6.91.

–

6.92.

–

6.93.

–

6.94.

–

6.95.

–

6.96.

–

6.97.

–

6.98.

–

6.99.

–

6.100.

–

6.101.

–

6.102.

–

6.103.

–

6.104.

–

6.105.

–

6.106.

–

6.107.

–

6.108.

–

6.109.

–

6.110.

–

6.111.

–

6.112.

–

6.113.

–

6.114.

–

6.115.

–

6.116.

–

6.117.

–

6.118.

–

6.119.

–

6.120.

–

7.

Circle Theorems

  • 1. Congruent Circles: Definition
  • 2. Circle Tangent-Types Illustrator
  • 3. Properties of Tangents Drawn to Circles (A)
  • 4. Properties of Tangents Drawn to Circles (B)
  • 5. Int+Ext Tangent Action!
  • 6. Diameters & Chords I (VA)
  • 7. Diameters & Chords I (VB)
  • 8. Congruent Chords: Quick Investigation
  • 9. Congruent Chords (I)
  • 10. Inscribed Angle Theorem (V1)
  • 11. Find that Angle! (GoGeometry Action 144)
  • 12. Proof Exercise: Inscribed Angle Theorem (Case 1)
  • 13. Proof Exercise: Inscribed Angle Theorem (Case 2)
  • 14. Proof Exercise: Inscribed Angle Theorem (Case 3)
  • 15. Inscribed Angle Theorem Dance: Take 2!
  • 16. Inscribed Angle Theorem: Corollary 1
  • 17. Where to Sit? (I)
  • 18. Thales' Theorem (VA)
  • 19. Thales' Theorem (VB)
  • 20. Inscribed Angle Intercepts Semicircle
  • 21. Cyclic Quadrilaterals (IAT: Corollary 3)
  • 22. Cyclic Quadrilateral: Proof Hint
  • 23. Cyclic Quadrilateral: Quick Warm Up (GoGeometry Action 149!)
  • 24. Angle Formed by Chord & Tangent (I)
  • 25. Angle Formed by Chord & Tangent (II)
  • 26. Angle Formed by 2 Chords (I)
  • 27. Angle Formed by 2 Chords (II)
  • 28. Crossing Chords: Proof Hint
  • 29. Angles from Secants and Tangents (V1)
  • 30. Angle From 2 Secants (V2)
  • 31. Secants: Proof Hint
  • 32. Congruent Chords Action (A)
  • 33. Equidistant Chord Action
  • 34. Butterfly Theorem Action!
  • 35. Not Your Everyday Chord & Tangent Theorem
  • 36. Cut-The-Knot Action (2) !!!
  • 37. Square, Tangent, and Two Congruent & Tangent Circles
  • 38. GoGeometry Action 1!
  • 39. GoGeometry Action 2!
  • 40. GoGeometry Action 3!
  • 41. GoGeometry Action 4!
  • 42. GoGeometry Action 5!
  • 43. Geometric Mean Illustration
  • 44. GoGeometry Action 6!
  • 45. GoGeometry Action 8!
  • 46. Cut-The-Knot Action (6)!
  • 47. GoGeometry Action 9!
  • 48. GoGeometry Action 13!
  • 49. GoGeometry Action 15!
  • 50. GoGeometry Action 16!
  • 51. Cut-The-Knot Action 7!
  • 52. GoGeometry Action 19!
  • 53. Thales Action + Sequel = GoGeometry Action 25!
  • 54. GoGeometry Action 26!
  • 55. GoGeometry Action 48!
  • 56. GoGeometry Action 49!
  • 57. GoGeometry Action 50!
  • 58. GoGeometry Action 52!
  • 59. 60-Degree Sector Surprise! (GoGeometry Action 55)
  • 60. Neusis Logic Action = GoGeometry Action 56!
  • 61. Quarter Circle Surprise!
  • 62. GoGeometry Action 59!
  • 63. GoGeometry Action 61!
  • 64. GoGeometry Action 62!
  • 65. GoGeometry Action 70!
  • 66. GoGeometry Action 72!
  • 67. Geometric Mean Madness! (GoGeometry Action 84)
  • 68. Mickey Mouse Theorem! (Cut-the-Knot Action 16)
  • 69. More Geometric Mean Action! (GoGeometry Action 86)
  • 70. GoGeometry Action 87!
  • 71. GoGeometry Action 92!
  • 72. A Stronger Conclusion!
  • 73. GoGeometry Action 100!
  • 74. GoGeometry Action 122!
  • 75. GoGeometry Action 128!
  • 76. GoGeometry Action 129!
  • 77. GoGeometry Action 131!
  • 78. GoGeometry Action 137!
  • 79. GoGeometry Action 141!
  • 80. GoGeometry Action 142!
  • 81. GoGeometry Action 148!
  • 82. Archimedes Insight = GoGeometry Action 150!
  • 83. More Archimedes Insight = GoGeometry Action 151!
  • 84. GoGeometry Action 154!
  • 85. GoGeometry Action 155!
  • 86. Archimedes Insight 3: GoGeometry Action 157!

7.1.

–

7.2.

–

7.3.

–

7.4.

–

7.5.

–

7.6.

–

7.7.

–

7.8.

–

7.9.

–

7.10.

–

7.11.

–

7.12.

–

7.13.

–

7.14.

–

7.15.

–

7.16.

–

7.17.

–

7.18.

–

7.19.

–

7.20.

–

7.21.

–

7.22.

–

7.23.

–

7.24.

–

7.25.

–

7.26.

–

7.27.

–

7.28.

–

7.29.

–

7.30.

–

7.31.

–

7.32.

–

7.33.

–

7.34.

–

7.35.

–

7.36.

–

7.37.

–

7.38.

–

7.39.

–

7.40.

–

7.41.

–

7.42.

–

7.43.

–

7.44.

–

7.45.

–

7.46.

–

7.47.

–

7.48.

–

7.49.

–

7.50.

–

7.51.

–

7.52.

–

7.53.

–

7.54.

–

7.55.

–

7.56.

–

7.57.

–

7.58.

–

7.59.

–

7.60.

–

7.61.

–

7.62.

–

7.63.

–

7.64.

–

7.65.

–

7.66.

–

7.67.

–

7.68.

–

7.69.

–

7.70.

–

7.71.

–

7.72.

–

7.73.

–

7.74.

–

7.75.

–

7.76.

–

7.77.

–

7.78.

–

7.79.

–

7.80.

–

7.81.

–

7.82.

–

7.83.

–

7.84.

–

7.85.

–

7.86.

–

8.

Geometric Constructions

  • 1. Isosceles Triangle (Quick Construction Technique) V1
  • 2. Equilateral Triangle Construction (Dynamic Illustration)
  • 3. Congruent Angle Construction Action!
  • 4. Angle Bisector Construction Action!
  • 5. Alternate Angle Bisector Construction
  • 6. Bizzare Trisection?
  • 7. Angle Trisection by Paper Folding!
  • 8. Bizarre Trisection (V2) = GoGeometry Action 145!
  • 9. Bizarre Trisection (V3) = GoGeometry Action 146!

8.1.

–

8.2.

–

8.3.

–

8.4.

–

8.5.

–

8.6.

–

8.7.

–

8.8.

–

8.9.

–

9.

Transformations

  • 1. Dilating a Point (Intro)
  • 2. Dilating a Line: HSG.SRT.A.1.A
  • 3. Dilating a Segment: HSG.SRT.A.1.B
  • 4. Properties of Dilations

9.1.

–

9.2.

–

9.3.

–

9.4.

–

10.

Area, Surface Area, Volume

  • 1. Parallelogram: Area
  • 2. Triangle Area Action! (V1)
  • 3. Area of a Triangle (Discovery)
  • 4. Trapezoid: Area (I)
  • 5. Area of a Trapezoid (Discovery)
  • 6. Scaling Figures: Effects on Areas
  • 7. SolveMyMaths Action 3A!
  • 8. SolveMyMaths Action 3B!
  • 9. SolveMyMaths Action 4!
  • 10. Right Circular Cylinder!

10.1.

–

10.2.

–

10.3.

–

10.4.

–

10.5.

–

10.6.

–

10.7.

–

10.8.

–

10.9.

–

10.10.

–

11.

Slope

  • 1. Parallel Consequence: Quick Reminder
  • 2. Parallel Consequence? (VA)
  • 3. Parallel Consequence? (VB)
  • 4. Parallel Lines in the Coordinate Plane (VA)
  • 5. Parallel Lines in the Coordinate Plane (VB)
  • 6. Perpendicular Lines in the CP: Quick Investigation
  • 7. Slope Triangle Rotation (VA)
  • 8. Slope Triangle Rotation (VB)

11.1.

–

11.2.

–

11.3.

–

11.4.

–

11.5.

–

11.6.

–

11.7.

–

11.8.

–

12.

Functions

  • 1. Functions: Input & Output (3)
  • 2. Functions: Input & Output (4)
  • 3. Half-Life Action!!!
  • 4. What Makes a Function Even?
  • 5. What Makes a Function Odd?
  • 6. Odd Functions (Another Look)
  • 7. Odd Functions: Another Look (Take 2)!
  • 8. Function Composition: Dynamic Illustrator (1)
  • 9. Periodic Function Action!
  • 10. Logarithmic Action!
  • 11. Logarithmic Action (2)!
  • 12. Logarithmic Action (3)!
  • 13. Logarithmic Action (4)!
  • 14. Logarithmic Action (4)! V2

12.1.

–

12.2.

–

12.3.

–

12.4.

–

12.5.

–

12.6.

–

12.7.

–

12.8.

–

12.9.

–

12.10.

–

12.11.

–

12.12.

–

12.13.

–

12.14.

–

13.

Vectors

  • 1. Vector Addition
  • 2. Vector Subtraction
  • 3. Scaling Vectors
  • 4. Vector Projections (1)

13.1.

–

13.2.

–

13.3.

–

13.4.

–

14.

Conic Sections: Definitions & Properties

  • 1. Parabola as a Locus
  • 2. Parabola: Cause and Effect
  • 3. Special Conic LR Action
  • 4. Parabola: Geometric Property (I)
  • 5. Parabola: Geometric Property (II)
  • 6. Another Parabola Theorem!
  • 7. Ellipse: Magic Sum = ?
  • 8. Ellipse: Magic Sum = ? (V2)
  • 9. Ellipse: Reflective Property
  • 10. Ellipse: Reflective Property
  • 11. Hyperbola: Difference = ?
  • 12. Hyperbola: Reflective Property

14.1.

–

14.2.

–

14.3.

–

14.4.

–

14.5.

–

14.6.

–

14.7.

–

14.8.

–

14.9.

–

14.10.

–

14.11.

–

14.12.

–

15.

Trigonometry

  • 1. Angle Orientations (Standard Position)
  • 2. Coterminal Angles Action!!!
  • 3. Special Angles in Standard Position: Intro (V1)
  • 4. Special Angles in Standard Position: Intro (V2)
  • 5. Special Angles in Standard Position: Intro (V2) - Part 2
  • 6. Special Angles in Standard Position: Intro (V3)
  • 7. Sine & Cosine Period Action (2)!
  • 8. Tangent Period Action (2)!
  • 9. Right Triangle Trigonometry: Intro
  • 10. Trigonometric Ratios (Right Triangle Context)
  • 11. What "CO" in COsine Means
  • 12. Law of Sines (& Area)
  • 13. Law of Cosines: Discovery
  • 14. Bizzare Trig Identity?
  • 15. 3 Squares Problem
  • 16. Polar Coordinates: Introduction

15.1.

–

15.2.

–

15.3.

–

15.4.

–

15.5.

–

15.6.

–

15.7.

–

15.8.

–

15.9.

–

15.10.

–

15.11.

–

15.12.

–

15.13.

–

15.14.

–

15.15.

–

15.16.

–

16.

Illustrations in 3-Space

  • 1. Non-Coplanar "Quad" Midpoints?
  • 2. Sphere Investigation
  • 3. Circular Paraboloid of Revolution
  • 4. Special Spheroid Action: Locus Illustration
  • 5. 3 Golden Rectangles Surprise!
  • 6. Dodecahedral Surprise!
  • 7. Dandelin Sphere Action!

16.1.

–

16.2.

–

16.3.

–

16.4.

–

16.5.

–

16.6.

–

16.7.

–

17.

Calculus Applets

  • 1. Derivative Function: Without Words
  • 2. Area Between Curves
  • 3. Mean Value Theorem (Without Words)

17.1.

–

17.2.

–

17.3.

–

18.

Cool But Not-Often-Seen Theorems

  • 1. Equilateral Action (1)!
  • 2. Equilateral Action (2)!
  • 3. Equilateral Action (3)!
  • 4. Napoleon Motion!
  • 5. Hexagonal Napoleon Theorem?
  • 6. Triangle to Conic?
  • 7. Square Problem
  • 8. Japanese Theorem for Cyclic Quadrilaterals
  • 9. Cyclic Quad Action (II)
  • 10. Equilateral Action (4)!
  • 11. Equilateral Triangle and Rectangle (Area Problem)
  • 12. Van Aubel Action!!!
  • 13. Finsler-Hadwiger Action!!!
  • 14. Morley Action!
  • 15. Quadrilateral Surprise!
  • 16. Cut-The-Knot Action !!!
  • 17. Cut-the-Knot-Action (3)!
  • 18. Cut-The-Knot-Action 11!
  • 19. GoGeometry Action 65!
  • 20. Hexagonal Surprise! (GoGeometry Action 85)
  • 21. GoGeometry Action 111!
  • 22. GoGeometry Action 123!
  • 23. Cut-The-Knot Action 17!
  • 24. From Triangle to WHAT?
  • 25. From Equiangular Hexagon to WHAT?
  • 26. GoGeometry Action 158!
  • 27. GoGeometry Action 159!
  • 28. Regular Decagon Magic!

18.1.

–

18.2.

–

18.3.

–

18.4.

–

18.5.

–

18.6.

–

18.7.

–

18.8.

–

18.9.

–

18.10.

–

18.11.

–

18.12.

–

18.13.

–

18.14.

–

18.15.

–

18.16.

–

18.17.

–

18.18.

–

18.19.

–

18.20.

–

18.21.

–

18.22.

–

18.23.

–

18.24.

–

18.25.

–

18.26.

–

18.27.

–

18.28.

–

Informazioni