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Demonstration of Incircle

Demonstration of Incircle
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2017-02-21 12:57
TriangleCircleGeometric Construction
This example demonstrates the dynamic changes of the incircle of a triangle. Key mathematical concepts include the incircle, angles, segments, and geometric transformations. Using GeoGebra commands such as Line, Circle, Incircle, and Rotate, it vividly shows the relationship between the incircle and the sides of the triangle. Suitable for students at the junior high school level and above to learn about geometric transformations and the concept of incircle.

GeoGebra Commands & Parameters

  1. Line(A, B) → d
  2. Slope(d) → a
  3. PointIn(c) → C
  4. Line(A, C) → e
  5. Line(B, C) → f
  6. Angle(B, A, C) → α
  7. Angle(A, C, B) → β
  8. Angle(C, B, A) → γ
  9. Segment(A, C) → g
  10. Segment(C, B) → h
  11. Segment(B, A) → i
  12. Min(list1) → j
  13. Circle(C, (0.2 * j)) → q
  14. Circle(A, (0.2 * j)) → p
  15. Circle(B, (0.2 * j)) → k
  16. Intersect(q, f, 2) → D
  17. Intersect(q, e, 1) → E
  18. Intersect(q, e, 2) → F
  19. Intersect(p, e, 2) → G
  20. Intersect(p, d, 1) → H
  21. Intersect(k, f, 2) → I_2
  22. Intersect(k, d, 2) → J
  23. Intersect(d, i) → K
  24. Intersect(d, i) → L
  25. Intersect(k, f, 1) → N
  26. Intersect(p, e, 1) → O
  27. Intersect(p, d, 2) → P
  28. Angle(D, C, E) → δ
  29. Angle(O, A, P) → ε
  30. Angle(J, B, I_2) → η
  31. Angle(G, A, H) → θ
  32. Incircle(A, B, C) → r
  33. Center(r) → I
  34. Line(C, I) → l
  35. Line(A, I) → n
  36. Line(B, I) → s
  37. Intersect(e, s) → R
  38. Intersect(d, l) → S
  39. Intersect(f, n) → T
  40. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  41. Rotate(D, δ / 2, C) → D'
  42. Angle(D, C, D') → κ
  43. Rotate(J, η / 2, B) → J'
  44. Angle(J, B, J') → λ
  45. Ray(C, D') → t
  46. Ray(B, J') → e_1
  47. Rotate(O, ε / 2, A) → O'
  48. Angle(O, A, O') → ι
  49. Ray(A, O') → d_1
  50. Segment(C, S) → f_1
  51. Segment(A, T) → g_1
  52. Segment(B, R) → h_1
  53. Intersect(q, f_1, 1) → U
  54. Intersect(q, f, 1) → V
  55. Intersect(p, g_1, 1) → W
  56. Intersect(k, h_1, 1) → Z
  57. CircleSector(C, E, U) → k_1
  58. CircleSector(A, P, W) → q_1
  59. CircleSector(B, I_2, Z) → p_1
  60. Dilate(C, If[0 < m ≤ 1, 1 - m, 0], S) → C'
  61. Segment(C, C') → i_1
  62. Rotate(k_1, If[1 < m ≤ 2, ((m - 1) * β / 2), β / 2], C) → r_1
  63. Dilate(B, If[2 < m ≤ 3, 3 - m, 0], R) → B'
  64. Segment(B, B') → j_1
  65. Rotate(p_1, If[3 < m ≤ 4, ((m - 3) * γ / 2), γ / 2], B) → s_1
  66. Dilate(A, If[4 < m ≤ 5, 5 - m, 0], T) → A'
  67. Segment(A, A') → l_1
  68. Rotate(q_1, If[5 < m ≤ 6, ((m - 5) * α / 2), α / 2], A) → t_1
  69. OrthogonalLine(I, f) → m_1
  70. OrthogonalLine(I, d) → n_1
  71. OrthogonalLine(I, e) → a_2
  72. Intersect(f, m_1) → M
  73. Intersect(d, n_1) → A_1
  74. Intersect(e, a_2) → B_1
  75. Segment(I, M) → b_2
  76. Circle(M, (0.2 * b_2)) → c_2
  77. Intersect(c_2, f, 1) → C_1
  78. Intersect(c_2, b_2, 1) → D_1
  79. Intersect(c_2, f, 2) → E_1
  80. CircleSector(M, E_1, D_1) → d_2
  81. Dilate(I, If[6 < m ≤ 7, 7 - m, 0], M) → Q'
  82. Segment(I, Q') → e_2
  83. Rotate(d_2, If[7.5 < m ≤ 8.5, ((m - 7.5) * π / 2), 1.5707963267948966], M) → f_2
  84. Segment(I, B_1) → g_2
  85. Segment(I, A_1) → h_2
  86. Circle(B_1, (0.2 * g_2)) → k_2
  87. Circle(A_1, (0.2 * h_2)) → p_2
  88. Intersect(k_2, e, 2) → F_1
  89. Intersect(k_2, g_2, 1) → G_1
  90. Intersect(k_2, e, 1) → H_1
  91. Intersect(p_2, d, 2) → I_1
  92. Intersect(p_2, h_2, 1) → J_1
  93. Intersect(p_2, d, 1) → K_1
  94. Dilate(I, If[8.5 < m ≤ 9.5, 9.5 - m, 0], B_1) → Q'_1
  95. Segment(I, Q'_1) → i_2
  96. CircleSector(B_1, H_1, G_1) → q_2
  97. CircleSector(A_1, I_1, J_1) → r_2
  98. Rotate(q_2, If[10 < m ≤ 11, ((m - 10) * π / 2), 1.5707963267948966], B_1) → s_2
  99. Dilate(I, If[11 < m ≤ 12, 12 - m, 0], A_1) → Q'_2
  100. Segment(I, Q'_2) → j_2
  101. Rotate(r_2, If[12.5 < m ≤ 13.5, ((m - 12.5) * π / 2), 1.5708], A_1) → t_2
  102. Angle(M, I, B_1) → ζ
  103. Angle(B_1, I, A_1) → μ
  104. Angle(A_1, I, M) → ν
  105. Rotate(I, If[14 < m ≤ 15, ((m - 14) * ζ), ζ], I) → Q'_3
  106. Rotate(Q', If[14 < m ≤ 15, ((m - 14) * ζ), ζ], I) → Q''
  107. Segment(Q'_3, Q'') → l_2
  108. Rotate(Q'_3, If[15 < m ≤ 16, ((m - 15) * μ), μ], I) → L_1
  109. Rotate(Q'', If[15 < m ≤ 16, ((m - 15) * μ), μ], I) → Q'''
  110. Segment(L_1, Q''') → m_2
  111. Rotate(A_1, If[16 < m ≤ 18, ((m - 16) * π), 6.28319], I) → M'
  112. Intersect(m_1, f) → Q
  113. Circle(I, Q) → d_3
  114. Line(A, B) → n_2
  115. CircleArc(I, A_1, M') → c_3
  116. Segment(I, C) → f_3
  117. Intersect(d_3, f_3) → M_1,
  118. Segment(A, M_1) → g_3
  119. Segment(B, M_1) → h_3
  120. Segment(B_1, M) → i_3
  121. Intersect(g_3, i_3) → N_1
  122. Intersect(h_3, i_3) → O_1
  123. Intersect(f_3, i_3) → P_1
  124. Dilate(I, If[20 < m ≤ 21, 21 - m, 0], M_1) → I'
  125. Segment(I, I') → j_3
  126. Dilate(M_1, If[21 < m ≤ 22, 22 - m, 0], C) → Q_1
  127. Segment(M_1, Q_1) → k_3
  128. Dilate(A, If[22 < m ≤ 23, 23 - m, 0], N_1) → A'_1
  129. Segment(A, A'_1) → l_3
  130. Dilate(N_1, If[23 < m ≤ 24, 24 - m, 0], M_1) → R_1
  131. Segment(N_1, R_1) → m_3
  132. Dilate(B, If[22 < m ≤ 23, 23 - m, 0], O_1) → B'_1
  133. Dilate(O_1, If[23 < m ≤ 24, 24 - m, 0], M_1) → S_1
  134. Segment(B, B'_1) → n_3
  135. Segment(O_1, S_1) → p_3
  136. Dilate(B_1, If[24 < m ≤ 25, 25 - m, 0], N_1) → T_1
  137. Segment(B_1, T_1) → q_3
  138. Dilate(N_1, If[25 < m ≤ 26, 26 - m, 0], P_1) → U_1
  139. Segment(N_1, U_1) → r_3
  140. Dilate(P_1, If[26 < m ≤ 27, 27 - m, 0], O_1) → V_1
  141. Segment(P_1, V_1) → s_3
  142. Dilate(O_1, If[27 < m ≤ 28, 28 - m, 0], M) → W_1
  143. Segment(O_1, W_1) → t_3
  144. Midpoint(B_1, O_1) → Z_1
  145. Rotate(B_1, If[28 < m ≤ 29, ((m - 28) * π), π], Z_1) → A_2
  146. Rotate(T_1, If[28 < m ≤ 29, ((m - 28) * π), π], Z_1) → B_2
  147. Segment(A_2, B_2) → a_3
  148. Midpoint(N_1, M) → C_2
  149. Rotate(N_1, If[29 < m ≤ 30, ((m - 29) * π), π], C_2) → D_2
  150. Rotate(U_1, If[29 < m ≤ 30, ((m - 29) * π), π], C_2) → E_2
  151. Segment(D_2, E_2) → b_3

AI Drawing Prompt Reference

Draw a triangle with its incircle. Connect the incenter to each vertex, creating three angle bisectors. At each vertex, draw a small circle centered at that vertex with radius equal to one-fifth of the shortest side length. These small circles intersect the triangle's sides, producing multiple points. From each vertex, draw a ray that bisects the angle formed by the small circle and the adjacent sides. Mark the points where the incircle touches the sides. Finally, connect these touch points to form an inner triangle, and label all relevant angles.
The prompt above was compiled from the original drawing commands and is for reference only.