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Locus of the pole (polar) from tangents to the incircle

Locus of the pole (polar) from tangents to the incircle
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2024-02-20 06:01
TriangleCircleCalculusGeometric ConstructionLocus
In collections Besondere Punkte/Special Points
This example demonstrates how the “pole” (direct pole point in the polar/polar-line sense) corresponding to a tangent line of a triangle’s incircle moves as the tangent varies, producing a geometric locus. Key concepts include: the incircle of a triangle, tangency, pole–polar relationships (dual geometry), and the locus method. By treating “tangent line → corresponding pole” as a parameterized mapping, the construction tracks the changing pole directly in the plane. Construction workflow (from the GeoGebra command set): TriangleCenter to define relevant triangle centers/auxiliary geometric points; Circle to represent the incircle; Tangent to build a tangent at a variable point; Point to represent the tangent point and the resulting pole; OrthogonalLine, Line, Intersect, and Distance to determine necessary auxiliary lines, intersection points, and distances; LineBisector when angle-bisector constraints are needed; and finally Locus to render the path of the pole. Suitable for: advanced high-school enrichment or introductory university courses on synthetic geometry, polar geometry, and locus visualization.

GeoGebra Commands & Parameters

  1. TriangleCenter(A, B, C, 3) → O
  2. Line(A, B) → f
  3. Line(B, C) → g
  4. Line(C, A) → h
  5. Circle(B, C, A) → c
  6. Point(c) → D
  7. Tangent(D, c) → i,
  8. OrthogonalLine(A, i) → j
  9. OrthogonalLine(C, i) → k
  10. OrthogonalLine(B, i) → l
  11. Intersect(i, l) → E
  12. Intersect(i, j) → F
  13. Intersect(i, k) → G_1
  14. OrthogonalLine(E, h) → m
  15. OrthogonalLine(F, g) → p
  16. OrthogonalLine(G_1, f) → n
  17. Intersect(m, p) → H_1
  18. TriangleCenter(A, B, C, 4) → H
  19. TriangleCenter(A, B, C, 1) → I
  20. TriangleCenter(A, B, C, 5) → N
  21. OrthogonalLine(P, g) → q_1
  22. Intersect(g, q_1) → W
  23. OrthogonalLine(P, f) → p_1
  24. Intersect(f, p_1) → U
  25. Circle(P, Distance[I, U]) → d
  26. Point(d) → K
  27. Tangent(K, d) → q, n_1
  28. OrthogonalLine(A, q) → r
  29. OrthogonalLine(C, q) → s
  30. OrthogonalLine(B, q) → t
  31. Intersect(r, q) → L
  32. Intersect(q, s) → M
  33. Intersect(t, q) → N_1
  34. OrthogonalLine(L, g) → b
  35. OrthogonalLine(M, f) → e
  36. OrthogonalLine(N_1, h) → f_1
  37. Intersect(e, f_1) → O_1
  38. TriangleCenter(A, B, C, 2) → G
  39. Locus(O_1, K) → loc1
  40. Locus(H_1, D) → loc2
  41. Point(c) → J
  42. Line(P, J) → g_1
  43. OrthogonalLine(A, g_1) → h_1
  44. OrthogonalLine(B, g_1) → i_1
  45. Intersect(g_1, h_1) → R
  46. Intersect(g_1, i_1) → S
  47. OrthogonalLine(S, h) → k_1
  48. OrthogonalLine(R, g) → m_1
  49. OrthogonalLine(C, g_1) → j_1
  50. Intersect(g_1, j_1) → Q
  51. OrthogonalLine(Q, f) → l_1
  52. Intersect(k_1, m_1) → T
  53. OrthogonalLine(P, h) → r_1
  54. Intersect(r_1, h) → V
  55. LineBisector(U, W) → s_1
  56. LineBisector(U, V) → t_1
  57. Distance(W, N) → distanceWN

AI Drawing Prompt Reference

Start with a triangle ABC. Draw its incircle and consider a varying tangent line to this incircle at a moving tangency point. For each such tangent, construct the corresponding pole point K using the pole–polar relationship: build the needed auxiliary perpendicular lines from the incircle tangent setup, intersect with the appropriate lines defined by the triangle, and use the resulting intersections to place the pole. Trace the locus of K as the tangent changes, producing the curve of the pole under the tangent variation. Optionally, also plot the locus of the related constructed point based on the same auxiliary construction involving D, to show the paired locus behavior.
The prompt above was compiled from the original drawing commands and is for reference only.