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Opposite Angles of Cyclic Quadrilaterals (with tangent-chord angles)

Opposite Angles of Cyclic Quadrilaterals (with tangent-chord angles)
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2019-12-24 22:59
CirclePolygonAngleGeometric Construction
This example demonstrates the property that opposite angles of a cyclic quadrilateral are supplementary, including angles formed by tangents and chords. Key concepts involve cyclic quadrilaterals, angle properties, tangent-chord angles. Construction uses commands like Circle, Point, Intersect, Tangent, CircleArc, Segment, Angle, CircleSector, Rotate. Suitable for high school geometry education to explore circle and quadrilateral relationships and angle properties.

GeoGebra Commands & Parameters

  1. Circle(O, 1) → c
  2. Point(c) → C
  3. Circle(C, 0.3) → q
  4. Intersect(c, q, 2) → P
  5. Point(c) → A
  6. Intersect(c, q, 1) → N
  7. Tangent(A, c) → j,
  8. CircleArc(O, N, A) → d
  9. Point(d) → D
  10. CircleArc(O, A, P) → e
  11. Point(e) → B
  12. Segment(C, D) → f
  13. Segment(D, A) → g
  14. Segment(A, B) → h
  15. Segment(B, C) → i
  16. Circle(A, 1.5) → p
  17. Intersect(p, j, 2) → F
  18. Intersect(p, j, 1) → E
  19. If(k ≤ 1, k, 1) → k1
  20. If(k ≤ 1, 0, 1 < k ≤ 2, k - 1, 1) → k2
  21. If(k ≤ 2, 0, 2 < k ≤ 3, k - 2, 1) → k3
  22. If(k ≤ 3, 0, 3 < k ≤ 4, k - 3, 1) → k4
  23. Segment(G, H) → l
  24. Segment(C, I) → m
  25. Angle(B, A, D) → α
  26. Intersect(q, f, 1) → J
  27. Intersect(q, i, 1) → K
  28. Intersect(q, m, 1) → L
  29. CircleSector(C, J, L) → s1
  30. CircleSector(C, L, K) → s2
  31. Rotate(Translate[s1, Vector[(k3 * Vector[C, A])]], ((-k3) * Angle[A, D, C]), M) → s1'
  32. Rotate(Translate[s2, Vector[(k3 * Vector[C, A])]], (k3 * Angle[C, B, A]), M) → s2'
  33. Angle(D, C, B) → r