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8.3 Inscribed Polygons and Semicircle Theorem Exercise

8.3 Inscribed Polygons and Semicircle Theorem Exercise
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2015-01-19 07:52
CirclePolygonAngleTrigonometryGeometric Construction
In collections MATHEMATICS and Geometry 9 : 8.3 Iscribed Polygons
This example demonstrates the application of the Inscribed Polygon and Semicircle Theorem by constructing specific angle and circle relationships to solve for angles. Core mathematical concepts include inscribed polygons, circle theorems, and trigonometric principles. GeoGebra commands used in construction are Intersect, Circle, Point, Tangent, Rotate, Angle, Ray, Polygon, CircumcircleArc, LineBisector, and Segment. Suitable for high school students learning geometric constructions and theorem proofs.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → O
  2. Circle(O, 10) → circle
  3. Point(circle) → A
  4. Tangent(A, circle) → a,
  5. Rotate(O, 270*°, A) → O'
  6. Angle(O', A, O) → ρ
  7. Ray(A, O) → b
  8. Intersect(circle, b, 2) → B
  9. Point(circle) → D
  10. Point(circle) → C
  11. Polygon(A, B, C) → polígono1, c, a_1, b_1
  12. Angle(C, B, A) → β
  13. Point(circle) → E
  14. CircumcircleArc(C, E, B) → d
  15. LineBisector(C, B) → e
  16. Point(e) → F
  17. Polygon(A, F, B) → polígono2, b_2, a_2, f
  18. Polygon(A, C, B) → polígono3, b_3, a_3, c_1
  19. Polygon(A, F, B) → polígono4, b_4, a_4, f_1
  20. Intersect(e, a_1) → G
  21. Segment(C, G) → g
  22. Segment(G, B) → h
  23. CircumcircleArc(C, E, F) → k
  24. CircumcircleArc(F, D, B) → p
  25. Angle(A, C, B) → ϕ
  26. Angle(A, F, B) → θ
  27. Angle(F, A, C) → α
  28. Angle(B, A, F) → δ
  29. Angle(F, A, C) → γ
  30. Angle(B, A, F) → ε
  31. CircumcircleArc(C, E, F) → q
  32. CircumcircleArc(F, D, B) → r
  33. Segment(C, A) → i
  34. Segment(A, F) → j
  35. Point(q) → H
  36. Segment(F, B) → l
  37. Segment(C, F) → m

AI Drawing Prompt Reference

Draw a circle with center O and radius 10, then take point A on the circle and draw the tangent a at A. Rotate O 90 degrees to get O', construct the angle bisector e of ∠O'AO, which intersects the circle at F. Construct polygons AFB and ACB, and finally connect the points and label the key angles and arcs.
The prompt above was compiled from the original drawing commands and is for reference only.