Geometry 14-15 Investigations

Andy Talmadge, Jan 18, 2015

A collection of student investigations into geometry ideas. All students were studying geometry during the 2014-15 school year.

Table of Contents

  1. Special Curves
    1. D.I.Y. Harriss Spiral
    2. Nine Point Circle
    3. Reuleaux Triangle
    4. Directions to Make the Harriss Spiral
    5. Ceva's Theorem
    6. Circle of Apollonius
    7. Nine Point Circle
    8. Spiral of Archimedes
    9. Conchoid of Nicomedes
    10. Golden Spiral
    11. Conchoid of Nicomedes
    12. Reuleaux Triangle
    13. The Euler Line
    14. Simson Line
    15. Nine Point Circle
    16. Bezier Curve Interactive Demonstration
    17. Orthic Triangle
    18. Bezier Curves
    19. Golden Spiral
    20. Conchoid of Nicomedes pt2
    21. OrthicTriangle
    22. The Bezier Curve
    23. Excenter / Excircle
    24. Excenter/Excircle v2
    25. Circle of Apollonius
    26. The Simson Line
    27. Euler Line FINAL
    28. Spiral of Archimedes
    29. Euler line
  2. Transformations
    1. Projectivity and Perspectivity
    2. Desargues' Theorem
    3. Menelaus Theorem
    4. Heron's Theorem
    5. Sierpinski Stages of Creation
    6. Inversion in a Cirlce
    7. Inversion in a Circle
    8. Herons Theorem
    9. Menelaus' Theorem
  3. Special Points in Triangles
    1. Gergonne Point
    2. DIY Lemoine Point
    3. The Miquel Point
    4. Symmedian Point
    5. Lemoine Point
    6. Br0card P0oints
    7. Brocard Points
    8. Miquel Point
    9. Isogonal Conjugates
    10. Gergonne Point
    11. Symmedian point
    12. Excenter/Excircle
    13. miquel point
    14. Symmedian Point
    15. Symmedian
    16. Lemoine Point
    17. Symmedian Point

1.

Special Curves

  • 1. D.I.Y. Harriss Spiral
  • 2. Nine Point Circle
  • 3. Reuleaux Triangle
  • 4. Directions to Make the Harriss Spiral
  • 5. Ceva's Theorem
  • 6. Circle of Apollonius
  • 7. Nine Point Circle
  • 8. Spiral of Archimedes
  • 9. Conchoid of Nicomedes
  • 10. Golden Spiral
  • 11. Conchoid of Nicomedes
  • 12. Reuleaux Triangle
  • 13. The Euler Line
  • 14. Simson Line
  • 15. Nine Point Circle
  • 16. Bezier Curve Interactive Demonstration
  • 17. Orthic Triangle
  • 18. Bezier Curves
  • 19. Golden Spiral
  • 20. Conchoid of Nicomedes pt2
  • 21. OrthicTriangle
  • 22. The Bezier Curve
  • 23. Excenter / Excircle
  • 24. Excenter/Excircle v2
  • 25. Circle of Apollonius
  • 26. The Simson Line
  • 27. Euler Line FINAL
  • 28. Spiral of Archimedes
  • 29. Euler line

1.1.

D.I.Y. Harriss Spiral

Explore the process of creating a Harriss Spiral as well as the relationships between rectangles created by the 'hrec' tool, using the animated example.

-Master the ability to make small sections of a Harriss Spiral using the instruction provided. -Pay attention to the ratio shared between a long and short side of rectangle. How does this relate to the Golden Spiral?
– Alea Zone

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1.6.

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1.7.

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1.9.

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1.10.

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1.11.

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1.12.

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1.13.

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1.14.

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1.15.

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1.16.

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1.17.

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1.18.

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1.19.

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1.20.

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1.21.

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1.22.

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1.23.

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1.24.

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1.25.

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1.26.

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1.27.

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1.28.

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1.29.

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2.

Transformations

  • 1. Projectivity and Perspectivity
  • 2. Desargues' Theorem
  • 3. Menelaus Theorem
  • 4. Heron's Theorem
  • 5. Sierpinski Stages of Creation
  • 6. Inversion in a Cirlce
  • 7. Inversion in a Circle
  • 8. Herons Theorem
  • 9. Menelaus' Theorem

2.1.

Projectivity and Perspectivity

This is a drawing on Geogebra of a perspectivity. In geometry, a perspectivity is the formation of an image (points on diagonal line) in a plane of a scene (points on straight line) viewed from a fixed point. The science of perspectivity uses them to make realistic images in the proper proportion. You can move any point (except F1, F2, G1, G2, I1, I2, H1, and H2) around and explore. The triangle point is the point of perspectivity, where you would be seeing the object from.
– Mignon Daly

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2.9.

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3.

Special Points in Triangles

  • 1. Gergonne Point
  • 2. DIY Lemoine Point
  • 3. The Miquel Point
  • 4. Symmedian Point
  • 5. Lemoine Point
  • 6. Br0card P0oints
  • 7. Brocard Points
  • 8. Miquel Point
  • 9. Isogonal Conjugates
  • 10. Gergonne Point
  • 11. Symmedian point
  • 12. Excenter/Excircle
  • 13. miquel point
  • 14. Symmedian Point
  • 15. Symmedian
  • 16. Lemoine Point
  • 17. Symmedian Point

3.1.

Gergonne Point

In this Geogebra sketch, a triangle is shown with a gergonne point. The gergonne point is the place where the lines from the vertices to the tangent points of the incircle intersect. The incenter point is also shown. In the sketch points A, B, and C are movable. When moving point A, B, or C the dimensions of the triangle will change along with the position of the gergonne point and the incenter.

When the triangle is an equilateral triangle the gergonne point and the incenter point are at the same possition.
– Jack Muscarello

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3.7.

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3.8.

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3.9.

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3.10.

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3.11.

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3.12.

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3.13.

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3.14.

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3.15.

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3.16.

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3.17.

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