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Locus of Centroids of Triangles Inscribed in Parabola with Constant Area

Locus of Centroids of Triangles Inscribed in Parabola with Constant Area
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2019-12-01 03:49
TriangleLocus
This example demonstrates the locus of centroids of triangles inscribed in a parabola with a constant area. Key math concepts include centroid, parabola, inscribed triangle, and constant area. Construction involves GeoGebra commands like Point, Intersect, Segment, Mirror, Polygon, Root, Tangent, OrthogonalLine, Locus, and Distance. Suitable for high school students and math enthusiasts exploring properties of parabolas and geometric loci.

GeoGebra Commands & Parameters

  1. Point(yAxis) → M
  2. Intersect(xAxis, yAxis) → O
  3. Segment(O, M) → h_0
  4. Mirror(B_1, h_0) → B'
  5. Polygon(B_1, B', O) → T_1, a_0, b, b'
  6. Point(f, Mod[3 / 2 - t, 1]) → A
  7. Root(x^(3) + ((4 * s^(2)) * x^(2)) + ((5 * s^(4)) * x) + (2 * (s^(6) - ((4 * s^(2)) * T_1) - T_1))) → F, I, J
  8. Tangent(A, f) → g,
  9. OrthogonalLine(A, g) → h_1
  10. Intersect(f, i, 2) → B
  11. Intersect(f, i, 1) → C
  12. Polygon(A, C, B) → T, d, a, e
  13. Locus(H, A) → lugar1
  14. Intersect(i, yAxis) → C_1
  15. Intersect(h_1, i) → K
  16. Distance(A, K) → distanciaAK
  17. Distance(B, C) → distanciaDE
  18. Segment(K, A) → h

AI Drawing Prompt Reference

Draw a parabola and inscribe a triangle with a constant area on it, showing the locus of the triangle's centroid. Start by plotting the intersection of the parabola with the y-axis, then construct the triangle using reflection and mirroring, find its vertices, and draw the centroid. Finally, trace the path of the centroid while displaying the triangle, the parabola, and related auxiliary lines.
The prompt above was compiled from the original drawing commands and is for reference only.