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Triangle Medians and Related Geometric Constructions

Triangle Medians and Related Geometric Constructions
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2018-02-26 17:27
TrianglePolygonAngleTransformationGeometric Construction
In collections Folding paper, doing geometry figures
① This example demonstrates the construction of medians in a triangle, along with multiple related triangles and quadrilaterals created through rotations and translations, illustrating properties such as the concurrence of medians at the centroid. ② Core mathematical concepts include triangle medians, midpoints, centroid, geometric transformations (rotation, translation), angle bisectors, and circles. ③ Construction details: using GeoGebra commands such as Polygon to create the triangle, Midpoint for midpoints, Segment for medians, Rotate for rotations, CircleArc for arcs, Line for lines, AngularBisector for angle bisectors, and Intersect for intersection points; objects include triangles, quadrilaterals, medians, angle bisectors, and transformed figures. ④ Suitable for middle school geometry teaching, providing an intuitive understanding of median properties, the centroid, and effects of geometric transformations, and can serve as dynamic exploration material.

GeoGebra Commands & Parameters

  1. Polygon(A, C, D, B) → vierhoek1, a, c, d, b
  2. Midpoint(b) → N
  3. Polygon(B, E, D) → driehoek1, d_1, b_2, e_1
  4. Rotate(D, (((s - 1) * 180))°, d_1) → D'
  5. Rotate(E, (((s - 1) * 180))°, d_1) → E'
  6. Rotate(B, (((s - 1) * 180))°, d_1) → B'
  7. Polygon(B', E', D') → driehoek1', d', b', e'
  8. Midpoint(d') → M
  9. Polygon(B, A, C, E) → vierhoek2, b_1, a_1, c_1, e
  10. Rotate(B, 180°, d_1) → B'_1
  11. Rotate(E, 180°, d_1) → E'_1
  12. Rotate(D, 180°, d_1) → D'_1
  13. Polygon(B'_1, E'_1, D'_1) → driehoek1'_1, d'_1, b'_1, e'_1
  14. Segment(B, E) → f
  15. Rotate(D, -90°, B) → DD
  16. CircleArc(B, DD, D) → g
  17. Segment(A, E) → h
  18. Rotate(C, -135°, E) → C'
  19. CircleArc(E, C', C, xOyPlane) → k
  20. Polygon(A, C, E) → driehoek2, e_2, a_2, c_2
  21. Rotate(A, (((s - 3) * 180))°, h) → A'
  22. Rotate(C, -180°, h) → C'_1
  23. Rotate(E, (((s - 3) * (-180)))°, h) → E'_2
  24. Polygon(A, C'_1, E) → driehoek2', p, q, t
  25. Polygon(D'_1, A, E) → driehoek3, e_3, d'_2, a_3
  26. Polygon(A, C, E, DD) → vierhoek3, i, j, l, m
  27. Line(D'_1, E) → n
  28. AngularBisector(A, B, E) → r
  29. Intersect(p, b'_1) → F
  30. Segment(F, E) → g_1
  31. Segment(DD, F) → f_1
  32. Midpoint(A, E) → BB
  33. Segment(B, BB) → h_1
  34. Segment(A, C'_1) → i_1
  35. Segment(DD, E) → j_1
  36. Intersect(p, g_1) → I
  37. Line(A, M) → k_1
  38. Intersect(k_1, d') → J
  39. Segment(A, M) → l_1
  40. Line(E, N) → m_1
  41. Intersect(m_1, b) → K
  42. Intersect(h_1, l_1) → L
  43. Segment(E, N) → n_1