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Proof of Angle at Center and Circumference Theorem

Proof of Angle at Center and Circumference Theorem
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2014-02-03 11:13
CircleAngleGeometric Construction
This example demonstrates the proof of the theorem stating that the angle at the center of a circle is twice the angle at the circumference subtended by the same arc. It involves core mathematical concepts like circles, angles, triangles, and the inscribed angle theorem. The construction uses GeoGebra commands and objects such as Circle, Point, Segment, AngularBisector, Intersect, CircumcircleArc, Angle, Ray, and Line. It is suitable for high school geometry teaching to help students visually understand the theorem.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Point(c) → D
  4. Point(c) → E
  5. Segment(C, E) → a
  6. Segment(E, D) → b
  7. AngularBisector(C, E, D) → d
  8. Intersect(c, d, 2) → F
  9. CircumcircleArc(C, F, D) → e
  10. Angle(C, E, D) → α
  11. Ray(E, C) → f
  12. Ray(E, D) → g
  13. Line(E, A) → h
  14. Segment(C, A) → i
  15. Segment(D, A) → j
  16. Intersect(c, h, 2) → G
  17. Angle(C, A, G) → β
  18. Angle(G, A, D) → γ
  19. Angle(C, E, G) → δ
  20. Angle(G, E, D) → ε

AI Drawing Prompt Reference

Draw a circle and select three points C, D, and E on it. Connect CE and ED, and then draw the angle bisector of angle CED, which intersects the circle at point F. Draw the arc EF. Measure angles CED, CAD, and CAG, and finally measure angles EAG and EAD.
The prompt above was compiled from the original drawing commands and is for reference only.