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Projection of Apex of Right Pyramid

Projection of Apex of Right Pyramid
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2017-01-07 11:22
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the projection of the apex of a right pyramid onto a plane, illustrating key concepts in solid geometry such as projections, conic sections, polygons, angles, midpoints, angle bisectors, and more. It utilizes GeoGebra commands like Intersect, Point, Cone, Circle, Pyramid, OrthogonalLine, Midpoint, Line, AngularBisector, Segment, and Angle to construct the geometric figures. Suitable for high school and advanced students studying solid geometry, analytical geometry, and geometric constructions, as well as for mathematical education purposes.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → circumcentre
  2. Point(zAxis) → V
  3. Cone(circumcentre, V, 1) → a, c, b
  4. Point(c) → A
  5. Point(c) → B
  6. Point(c) → C
  7. Circle(A, C, B) → e
  8. Pyramid(C, A, B, V) → d, faceABC, edgeBV, faceACV, faceABV, faceBCV, edgeAC, edgeAB, edgeBC, edgeCV, edgeAV
  9. OrthogonalLine(B, edgeAC, xOyPlane) → f
  10. OrthogonalLine(A, edgeBC, xOyPlane) → g
  11. Intersect(f, g) → orthocentre
  12. Midpoint(edgeBC) → D
  13. Midpoint(edgeAC) → E
  14. Line(B, E) → h
  15. Line(A, D) → i
  16. Intersect(h, i) → centroid
  17. AngularBisector(edgeAC, edgeAB) → j, k
  18. AngularBisector(edgeAC, edgeBC) → l, m
  19. Intersect(k, m) → incentre
  20. Intersect(yAxis, zAxis) → V'
  21. Segment(V', V) → p
  22. Segment(V', A) → q
  23. Angle(V, V', A) → α
  24. OrthogonalLine(C, edgeAV, space) → s
  25. OrthogonalLine(B, edgeAV, space) → t