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Cone Section and Isosceles Triangle

Cone Section and Isosceles Triangle
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2015-08-10 20:43
TrigonometrySolid GeometryGeometric Construction
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
This example demonstrates the intersection of a cone with a plane to form a section, focusing on the generation of isosceles triangles. Key mathematical concepts include cone surfaces, section lines, properties of isosceles triangles, and the geometric meaning of centroids. GeoGebra commands used in construction include Polygon, PointIn, Translate, Rotate, Cone, Plane, IntersectPath, OrthogonalLine, Ends, Mirror, and QuadricSide. Suitable for high school solid geometry teaching to understand the section properties of spatial geometric bodies and the determination of isosceles triangles.

GeoGebra Commands & Parameters

  1. Polygon(B + ((-2), (-2)), B + (2, (-2)), B + (2, 2), B + ((-2), 2)) → poly3, e, f, g, i
  2. PointIn(poly3) → C
  3. Translate((0, 0, 0), Vector[B, C]) → A
  4. Polygon(A + ((-4), (-4)), A + (4, (-4)), A + (4, 4), A + ((-4), 4)) → poly1, a, b, c, d
  5. Rotate(Rotate[Translate[poly1, Vector[A, A + (0, 0, h)]], (90° * s), A + (0, 0, h), yAxis], (90° * t), A, zAxis) → poly2
  6. Rotate(D, 120*°, (0, 0, 0)) → E
  7. Cone((0, 0, 0), (0, 0, 4), 2) → m, e_1, o
  8. Polygon(D, E, 3, xOyPlane) → poly4, j, k, l, F
  9. PointIn(poly2) → G
  10. PointIn(poly2) → H
  11. PointIn(poly2) → I
  12. Plane(G, I, H) → n
  13. IntersectPath(n, m) → poly6
  14. OrthogonalLine(A, n) → d_1
  15. Ends(m) → g_1, g_1_1
  16. Mirror(m, T) → m'
  17. QuadricSide(m') → o'
  18. IntersectPath(n, m') → f_1
  19. Ends(m') → h_1, h_1_1

AI Drawing Prompt Reference

Create a cone with its vertex at the origin, axis along the z‑axis, height 4 and base radius 2. Draw two squares of different sizes in the horizontal plane, translate the larger square upward and rotate it about the y‑ and z‑axes to obtain an inclined plane. Choose three points on this plane to define a cutting plane that intersects the cone, producing a conic section curve. On the curve pick two points and join them with the cone vertex to form an isosceles triangle, draw the perpendicular from the vertex to the cutting plane, and reflect the cone across a point to show the symmetric opposite section.
The prompt above was compiled from the original drawing commands and is for reference only.