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Pythagorean Snail

Pythagorean Snail
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2022-06-22 20:02
TriangleCircleCoordinate Geometry
This example demonstrates the construction and properties of a Pythagorean snail, dynamically generating triangles and circles to illustrate its geometric patterns. Core mathematical concepts include the Pythagorean theorem, triangles, circles, coordinate geometry, and differential equations. Construction highlights from the GeoGebra command summary involve points, polygons, circles, orthogonal lines, intersections, and Archimedean spirals. Suitable for high school and above learners to understand geometric constructions, parametric equations, and dynamic geometric phenomena.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(yAxis) → B
  3. Point(yAxis) → C
  4. Polygon(A, B, C) → t1, c, a, b
  5. Circle(B, 1) → d
  6. OrthogonalLine(B, c) → f
  7. Intersect(d, f, 1) → D
  8. Polygon(D, B, A) → t2, a_1, d_1, b_1
  9. Archimedes(A, D) → e, t3, E
  10. Archimedes(A, E) → g, t4, F
  11. Archimedes(A, F) → h, t5, G
  12. Archimedes(A, G) → k, t6, H
  13. Archimedes(A, H) → p, t7, I
  14. Archimedes(A, I) → q, t8, J
  15. Archimedes(A, J) → r, t9, K
  16. Archimedes(A, K) → s, t10, L
  17. Archimedes(A, L) → t, t11, M
  18. Archimedes(A, M) → c_1, t12, N
  19. Archimedes(A, N) → e_1, t13, O
  20. Archimedes(A, O) → f_1, t14, P
  21. Archimedes(A, P) → g_1, t15, Q
  22. Archimedes(A, Q) → h_1, t16, R
  23. Archimedes(A, R) → k_1, t17, S
  24. Archimedes(A, S) → p_1, t18, T
  25. Archimedes(A, T) → q_1, t19, U

AI Drawing Prompt Reference

On a coordinate plane, choose three points on the axes to form a right triangle. From the middle vertex, draw the unit circle. Construct a line through the same vertex that is perpendicular to the line determined by the triangle, and find its intersection with the unit circle. Starting at one fixed triangle vertex, generate a sequence of Archimedean spiral curves by repeatedly using the previous key point and the newly obtained intersection point, extending the spiral outward step by step. The resulting figure should display the snail-like spiral winding, together with the underlying right-triangle and unit-circle relationships.
The prompt above was compiled from the original drawing commands and is for reference only.