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Spherical Circumcenter and Circumcircle of a Triangle

Spherical Circumcenter and Circumcircle of a Triangle
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2014-07-02 18:51
TriangleCircleCoordinate GeometryGeometric Construction
In collections Spherical Geometry Ideas
This example demonstrates the circumcenter and circumcircle of a triangle in Euclidean geometry. The circumcenter is the point where the perpendicular bisectors of the triangle's sides intersect, and it is equidistant from the triangle's vertices. Using GeoGebra commands such as Sphere, PointIn, Plane, IntersectPath, etc., the circumcircle of a triangle can be constructed. Suitable for students at the junior high school level and above for geometric construction.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, 0), 1) → a
  2. PointIn(a) → A
  3. PointIn(a) → B
  4. PointIn(a) → C
  5. Plane(A, O_1, B) → b
  6. Plane(B, O_1, C) → c
  7. Plane(C, O_1, A) → d
  8. IntersectPath(b, a) → e
  9. IntersectPath(c, a) → f
  10. IntersectPath(d, a) → g
  11. OrthogonalLine(O_1, b) → h
  12. OrthogonalLine(O_1, d) → i
  13. OrthogonalLine(O_1, c) → j
  14. Midpoint(A, C) → D
  15. Midpoint(C, B) → E
  16. Midpoint(B, A) → F
  17. Plane(D, i) → k
  18. IntersectPath(k, a) → p
  19. Plane(F, h) → l
  20. IntersectPath(l, a) → q
  21. Plane(E, j) → m
  22. IntersectPath(m, a) → r
  23. Intersect(p, r, 1) → O
  24. Line(O, O_1) → n
  25. Circle(n, B) → s

AI Drawing Prompt Reference

Draw the circumcenter and circumcircle of a triangle. Start by drawing a unit sphere in 3D space and taking three points A, B, and C on the sphere as the vertices of the triangle. Then construct the perpendicular bisectors of the triangle's sides, and their intersection is the circumcenter O. Finally, draw a circle with O as the center and any vertex of the triangle as the radius, which is the circumcircle of the triangle.
The prompt above was compiled from the original drawing commands and is for reference only.