Special Property of Equilateral Triangle

GeoGebra Commands & Parameters

1. Polygon(A, B, 3) → poly1, f, g, h, C
2. Segment(A, B) → i
3. Segment(B, C) → j
4. Segment(C, A) → k
5. Centroid(poly1) → D
6. OrthogonalLine(D, g) → l
7. OrthogonalLine(D, i) → m
8. Intersect(l, g) → E
9. Circle(D, E) → c
10. Angle(B, A, C) → α
11. Rotate(B, 30°, A) → B'
12. Angle(B, A, B') → β
13. Line(A, B') → n
14. Intersect(c, n, 1) → F
15. Midpoint(A, F) → G
16. Tangent(F, c) → p,
17. Intersect(p, i) → H
18. Intersect(p, h) → I
19. Polygon(A, H, I) → poly2, i_1, a, h_1
20. Centroid(poly2) → J
21. Circle(J, F) → d
22. Intersect(c, m, 2) → K
23. Tangent(K, c) → q,
24. Intersect(q, g) → L
25. Intersect(q, h) → M
26. Polygon(M, L, C) → poly3, c_1, m_1, l_1
27. Centroid(poly3) → N
28. Circle(N, K) → e
29. Line(B, D) → r
30. Intersect(h, k) → O
31. Intersect(r, h) → P
32. Intersect(c, r) → Q, R
33. Tangent(Q, c) → s,
34. Intersect(s, g) → S
35. Intersect(s, i) → T
36. Polygon(T, B, S) → poly4, s_1, t, b
37. Centroid(poly4) → U
38. Circle(U, Q) → d_1
39. Point(cc) → V
40. Tangent(V, ee) → f_1, g_1
41. Tangent(V, dd) → j_1, k_1
42. Tangent(V, ddd) → n_1, p_1
43. Intersect(e, ee, 1) → W
44. Intersect(e, ee) → Z, , ,
45. Intersect(ee, g_1) → A_1
46. Intersect(dd, j_1) → B_1
47. Intersect(ddd, n_1) → C_1
48. Segment(V, A_1) → q_1
49. Segment(V, B_1) → r_1
50. Segment(V, C_1) → t_1
51. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
52. Segment(A, A') → a_1
53. Rotate(A, If[1 < o ≤ 2, ((o - 1) * π / 3), π / 3], A) → A'_1
54. Rotate(A', If[1 < o ≤ 2, ((o - 1) * π / 3), π / 3], A) → A''
55. Segment(A'_1, A'') → b_1
56. Rotate(A, If[2 < o ≤ 3, ((-(o - 2)) * π / 3), ((-π)) / 3], B) → A'_2
57. Rotate(A', If[2 < o ≤ 3, ((-(o - 2)) * π / 3), ((-π)) / 3], B) → A''_1
58. Segment(A'_2, A''_1) → e_1
59. Max(list1) → u
60. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], A_1) → V'
61. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], C_1) → V'_2
62. Circle(V, q_1) → c_2
63. Intersect(c_2, c, 1) → D_1
64. Intersect(c_2, c) → E_1, F_1, ,
65. Intersect(c_2, r_1) → G_1,
66. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], G_1) → V'_1
67. Dilate(G_1, If[5 < o ≤ 6, 6 - o, 0], B_1) → H_1
68. Segment(V, V') → f_2
69. Segment(V, V'_1) → g_2
70. Segment(V, V'_2) → h_2
71. Segment(G_1, H_1) → i_2
72. Angle(A_1, V, B_1) → γ
73. Rotate(V, If[6 < o ≤ 7, ((o - 6) * γ), γ], V) → V'_3
74. Rotate(V', If[6 < o ≤ 7, ((o - 6) * γ), γ], V) → V''
75. Segment(V'_3, V'') → j_2
76. Dilate(V, If[7 < o ≤ 8, 8 - o, 0], G_1) → V'_4
77. Vector(V, V'_4) → v
78. Translate(V, v) → V'_5
79. Translate(V'_2, v) → I_1
80. Segment(V'_5, I_1) → k_2
81. Angle(B_1, V, C_1) → δ
82. Rotate(V'_5, If[8 < o ≤ 9, ((-(o - 8)) * δ), (-δ)], V'') → J_1
83. Rotate(I_1, If[8 < o ≤ 9, ((-(o - 8)) * δ), (-δ)], V'') → K_1
84. Segment(J_1, K_1) → l_2
85. Circle(V, r_1) → d_2
86. Intersect(d_2, t_1) → L_1,
87. Segment(V, V') → m_2
88. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], B_1) → V'_6
89. Segment(V, V'_6) → n_2
90. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], L_1) → V'_7
91. Segment(V, V'_7) → p_2
92. Dilate(L_1, If[5 < o ≤ 6, 6 - o, 0], C_1) → M_1
93. Segment(L_1, M_1) → q_2
94. Angle(B_1, V, L_1) → ε
95. Rotate(V, If[6 < o ≤ 7, ((o - 6) * ε), ε], V) → V'_8
96. Rotate(V'_6, If[6 < o ≤ 7, ((o - 6) * ε), ε], V) → N_1
97. Segment(V'_8, N_1) → r_2
98. Dilate(V, If[7 < o ≤ 8, 8 - o, 0], N_1) → V'_9
99. Vector(V, V'_9) → w
100. Translate(V, w) → V'_{10}
101. Translate(V', w) → V''_1
102. Segment(V'_{10}, V''_1) → s_2
103. Angle(N_1, V, A_1) → ζ
104. Rotate(V'_{10}, If[8 < o ≤ 9, ((-(o - 8)) * ζ), (-ζ)], N_1) → O_1
105. Rotate(V''_1, If[8 < o ≤ 9, ((-(o - 8)) * ζ), (-ζ)], N_1) → P_1
106. Segment(O_1, P_1) → t_2
107. Circle(V, t_1) → e_2
108. Intersect(e_2, q_1) → Q_1,
109. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], Q_1) → V'_{11}
110. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], B_1) → V'_{12}
111. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], C_1) → V'_{13}
112. Segment(V, V'_{11}) → a_2
113. Segment(V, V'_{12}) → b_2
114. Segment(V, V'_{13}) → f_3
115. Dilate(Q_1, If[5 < o ≤ 6, 6 - o, 0], A_1) → R_1
116. Segment(Q_1, R_1) → g_3
117. Angle(C_1, V, Q_1) → η
118. Rotate(V, If[6 < o ≤ 7, ((o - 6) * η), η], V) → V'_{14}
119. Rotate(V'_{13}, If[6 < o ≤ 7, ((o - 6) * η), η], V) → S_1
120. Segment(V'_{14}, S_1) → h_3
121. Dilate(V, If[7 < o ≤ 8, 8 - o, 0], S_1) → V'_{15}
122. Vector(V, V'_{15}) → u_1
123. Translate(V, u_1) → V'_{16}
124. Translate(V'_{12}, u_1) → T_1
125. Segment(V'_{16}, T_1) → i_3
126. Angle(A_1, V, B_1) → θ
127. Rotate(V'_{16}, If[8 < o ≤ 9, ((-(o - 8)) * θ), (-θ)], S_1) → U_1
128. Rotate(T_1, If[8 < o ≤ 9, ((-(o - 8)) * θ), (-θ)], S_1) → V_1
129. Segment(U_1, V_1) → j_3