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Another Special Theorem: Part 1 (V2) - Angle Bisector Property Exploration

Another Special Theorem: Part 1 (V2) - Angle Bisector Property Exploration
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2016-06-20 22:55
TriangleAngleTransformationGeometric ConstructionLocus
In collections Congruence (CO) Cluster Congruence (Vol 2) CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Gail LeBlancCCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2
This interactive GeoGebra example guides students to discover and verify a geometric theorem: if a point is equidistant from the sides of an angle, then it lies on the angle bisector. Designed with less scaffolding, it encourages independent exploration and includes key guiding questions. Core mathematical concepts include angle bisectors, distance from a point to a line, perpendicularity, and theorem discovery. Construction highlights: using GeoGebra commands such as Rotate, Ray, Angle, OrthogonalLine, Intersect, Distance, Min, Circle, CircleSector, Dilate, and Segment to build the angle, perpendiculars, and measurements. Suitable for middle or high school geometry, especially for inquiry-based learning, teacher demonstrations, or student self-study.

GeoGebra Commands & Parameters

  1. Rotate(A, α, B) → A'
  2. Angle(A, B, A') → β
  3. Ray(B, A) → f
  4. Ray(B, A') → g
  5. Rotate(A, β / 2, B) → A'_1
  6. Angle(A, B, A'_1) → γ
  7. Ray(B, A'_1) → h
  8. Point(h) → C
  9. OrthogonalLine(C, f) → i
  10. OrthogonalLine(C, g) → j
  11. Intersect(j, g) → D
  12. Intersect(i, f) → E
  13. Distance(E, B) → a
  14. Distance(E, C) → b
  15. Min(a, b) → c
  16. Circle(E, (0.2 * c)) → d
  17. Circle(D, (0.2 * c)) → e
  18. Intersect(e, j, 1) → F
  19. Intersect(d, i, 2) → H
  20. Intersect(d, f, 2) → G
  21. Intersect(e, g, 2) → I
  22. CircleSector(E, G, H) → k
  23. CircleSector(D, F, I) → p
  24. Dilate(C, If[0 < l ≤ 1, 1 - l, 0], D) → C'_1
  25. Dilate(C, If[1 < l ≤ 2, 2 - l, 0], E) → C'
  26. Segment(C, C'_1) → m
  27. Segment(C, C') → n
  28. Rotate(p, If[2 < l ≤ 3, ((-(l - 2)) * π / 2), ((-π)) / 2], D) → p'
  29. Rotate(k, If[3 < l ≤ 4, ((l - 3) * π / 2), π / 2], E) → k'
  30. Angle(D, C, E) → δ
  31. Rotate(C, If[4 < l ≤ 5, ((l - 4) * δ), δ], C) → C'_2
  32. Rotate(C'_1, If[4 < l ≤ 5, ((l - 4) * δ), δ], C) → J
  33. Segment(C'_2, J) → m'
  34. Circle(B, (0.3 * c)) → q
  35. Intersect(q, f, 1) → L
  36. Intersect(q, h, 1) → M
  37. CircleSector(B, L, M) → r
  38. Rotate(r, If[5.75 < l ≤ 6.75, ((l - 5.75) * α / 2), α / 2], B) → r'

AI Drawing Prompt Reference

Draw a geometry figure for exploring the angle-bisector property: construct an angle and its bisector, place a point on the bisector, and drop perpendiculars from that point to both sides, marking the feet. Show the two perpendicular segments as distances to the sides, then draw small circles centered at the feet and use the resulting sectors to highlight equal distances. Add scaled positions of the point along the perpendicular directions, connect them with segments, rotate the two foot-centered sectors, and finally draw a small sector at the angle vertex between a side and the bisector, rotating it by half the angle to suggest the bisector relationship.
The prompt above was compiled from the original drawing commands and is for reference only.