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The 3rd Set of Nets: Rotation, Orthocenter & Pythagorean Links

The 3rd Set of Nets: Rotation, Orthocenter & Pythagorean Links
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2023-11-11 15:34
TrianglePolygonTrigonometryTransformationGeometric Construction
In collections How many nets of a cuboid are there?
This GeoGebra example demonstrates how to generate a “net” style geometric configuration by applying rotations so that special points (notably an orthocenter-related structure) and the resulting relationships remain visually trackable under transformation. Key concepts include rotation (Rotation), the Pythagorean idea (Pythagoras/Pythagorean Theorem), and orthocenter geometry (Orthocenter) within a triangle, with LCM/GCD used to synchronize or parameterize lengths/scales in the construction. Construction-wise, it uses Point and Polygon to define the vertices and polygonal regions, and Rotate to create rotated copies of points/parts (e.g., corresponding sets like q2, q3, q4 and their primed counterparts such as q2', q3', q4'). Additional adjustable parameters (t, t', l, w, h) and control elements (button1, button2) support switching between sets and animating the transformation to reveal invariant structure. Suitable for secondary-level geometry exploration, dynamic-geometry demonstrations, and students learning how transformations expose classic triangle properties.

GeoGebra Commands & Parameters

  1. Point(Ray[(1, 0), (3, 0)]) → A
  2. Point(Ray[A + (1, 0), A + (2, 0)]) → C
  3. Point(Ray[A + (0, 1), A + (0, 2)]) → B
  4. Polygon({(0, 0), A, B, (0, y(B))}) → q1
  5. Polygon({A, C, (x(C), y(B)), B}) → q2
  6. Polygon({B, B + (0, h), B + ((-l), h), B + ((-l), 0)}) → q4
  7. Polygon({C, C + (l, 0), C + (l, w), C + (0, w)}) → q5
  8. Polygon({C + (l, 0), C + (l + h, 0), C + (l + h, w), C + (l, w)}) → q6
  9. Rotate(q4, (90° * t), Line[(0, w), B]) → q4'
  10. Rotate(q2, (90° * t), Line[B, A]) → q2'
  11. Rotate(Rotate[q5, (90° * t), Line[C + (0, w), C]], (90° * t), Line[B, A]) → q5'
  12. Rotate(Rotate[Rotate[q6, (90° * t), Line[C + (l, w), C + (l, 0)]], (90° * t), Line[C + (0, w), C]], (90° * t), Line[B, A]) → q6'_{1}
  13. Polygon({C, C + (0, (-h)), C + (l, (-h)), C + (l, 0)}) → q3
  14. Rotate(Rotate[Rotate[q3, (90° * t), Line[C + (l, 0), C]], (90° * t), Line[C + (0, w), C]], (90° * t), Line[B, A]) → q3'

AI Drawing Prompt Reference

This is a 'net'-style geometric configuration generated by rotation, containing a triangle and multiple polygons. Through rotation, it demonstrates orthocenter-related structures and Pythagorean theorem relationships. The construction order is to first define points A, B, C, then draw polygons q1, q2, q4, q5, q6, q3, and finally rotate these polygons to obtain q4', q2', q5', q6'_{1}, q3'. Parameters t, t', l, w, h control rotation angles and scales.
The prompt above was compiled from the original drawing commands and is for reference only.