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Spherical Triangle: A Quadrant Triangle

Spherical Triangle: A Quadrant Triangle
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2014-07-15 08:31
TrigonometrySolid GeometryGeometric Construction
In collections Spherical Geometry Ideas
This example demonstrates a QUADRANT TRIANGLE on a sphere where one side, AB, is a quadrant (90 degrees). It involves core concepts of spherical geometry, triangle properties, and angle measurement. The construction uses GeoGebra commands like Sphere, Intersect, CurveCartesian, PointIn, Plane, OrthogonalLine, Segment, Angle, and OrthogonalPlane. Suitable for high school and above students learning about spherical geometry and special triangle properties.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), 1) → a
  2. Intersect(yAxis, a, 1) → A
  3. Intersect(xAxis, a, 2) → B
  4. CurveCartesian(cos(t), sin(t), 0, t, -90*°, 0*°) → b
  5. PointIn(a) → C
  6. Plane(C, D, A) → c
  7. OrthogonalLine(D, c) → d
  8. Plane(A, d) → e
  9. OrthogonalLine(D, e) → f
  10. Intersect(f, a, 2) → F
  11. CurveCartesian((x(A) * cos(t)) + (x(F) * sin(t)), (y(A) * cos(t)) + (y(F) * sin(t)), (z(A) * cos(t)) + (z(F) * sin(t)), t, 0, Angle[A, D, C]) → g
  12. Plane(C, D, B) → h
  13. OrthogonalLine(D, h) → i
  14. Plane(B, i) → j
  15. OrthogonalLine(D, j) → k
  16. Intersect(k, a, 2) → E
  17. Segment(A, D) → m
  18. Segment(D, B) → n
  19. Angle(A, D, B) → α
  20. Angle(C, D, B) → β
  21. CurveCartesian((x(B) * cos(t)) + (x(E) * sin(t)), (y(B) * cos(t)) + (y(E) * sin(t)), (z(B) * cos(t)) + (z(E) * sin(t)), t, 0, β) → l
  22. Segment(D, C) → p
  23. OrthogonalPlane(C, p) → o

AI Drawing Prompt Reference

Draw a spherical triangle with one side AB as a quarter-circle arc on the sphere. Start by drawing a unit sphere centered at O, then find points A and B on the x-axis and y-axis, respectively. Next, draw a quarter-circle arc starting from point C on the sphere to form triangle ABC. Construct a plane c through C and D, and draw a line d perpendicular to CD on this plane. Then, draw a line f from D that intersects the sphere, finding the intersection point F. Draw a curve g from A through D that intersects the sphere. Finally, draw a curve l from B through D that intersects the sphere.
The prompt above was compiled from the original drawing commands and is for reference only.