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Mikko's Hexagon Theorem in Triangle

Mikko's Hexagon Theorem in Triangle
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2025-08-22 01:10
TrianglePolygonGeometric Construction
In collections Besondere Punkte/Special Points
This example demonstrates the construction of a hexagon in a triangle and applies related theorems. Key mathematical concepts include geometric transformations, polygon properties, and inscribed triangles. Construction involves GeoGebra commands such as Polygon, Segment, Intersect, Rotate, and Line. Suitable for middle or high school mathematics to understand geometric transformations and figure construction.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → kolmio, c, a, b
  2. Segment(E, G) → f
  3. Segment(G, I) → g
  4. Segment(I, E) → h
  5. Segment(D, F) → i
  6. Segment(F, H) → j
  7. Segment(H, D) → k
  8. Intersect(h, i) → J
  9. Intersect(f, i) → K
  10. Intersect(f, j) → L
  11. Intersect(g, j) → M
  12. Intersect(g, k) → N
  13. Intersect(h, k) → O
  14. Polygon(K, L, M, N, O, J) → kuusik, k_1, l, m, n, o, j_1
  15. Polygon(I, O, H, N, D, J, E, K, F, L, G, M) → tahti, i_1, m_1, h_1, n_1, o_1, d_1, j_2, e_1, k_2, f_1, l_1, g_1
  16. Polygon(A_{1}, B_{1}, C_{1}) → kolmio_{1}, c_{1}, a_{1}, b_{1}
  17. Polygon(A_{2}, B_{2}, C_{2}) → kolmio_{2}, c_{2}, a_{2}, b_{2}
  18. Segment(E_{1}, G_{1}) → f_{2}
  19. Segment(G_{1}, I_{1}) → g_{2}
  20. Segment(I_{1}, E_{1}) → h_{2}
  21. Segment(D_{1}, F_{1}) → i_{2}
  22. Segment(F_{1}, H_{1}) → j_{3}
  23. Segment(H_{1}, D_{1}) → k_{3}
  24. Intersect(h_{2}, i_{2}) → J_{1}
  25. Intersect(f_{2}, i_{2}) → K_{1}
  26. Intersect(f_{2}, j_{3}) → L_{1}
  27. Intersect(g_{2}, j_{3}) → M_{1}
  28. Intersect(g_{2}, k_{3}) → N_{1}
  29. Intersect(h_{2}, k_{3}) → O_{1}
  30. Polygon(K_{1}, L_{1}, M_{1}, N_{1}, O_{1}, J_{1}) → kuusik_{1}, k_{4}, l_{2}, m_{2}, n_{2}, o_{2}, j_{4}
  31. Polygon(I_{1}, O_{1}, H_{1}, N_{1}, D_{1}, J_{1}, E_{1}, K_{1}, F_{1}, L_{1}, G_{1}, M_{1}) → tahti_{1}, i_{3}, m_{3}, h_{3}, n_{3}, o_{3}, d_{2}, j_{5}, e_{2}, k_{5}, f_{3}, l_{3}, g_{3}
  32. Polygon(A_{3}, B_{3}, C_{3}) → kolmio_{3}, c_{3}, a_{3}, b_{3}
  33. Segment(E_{2}, G_{2}) → f_{4}
  34. Segment(G_{2}, I_{2}) → g_{4}
  35. Segment(I_{2}, E_{2}) → h_{4}
  36. Segment(D_{2}, F_{2}) → i_{4}
  37. Segment(F_{2}, H_{2}) → j_{6}
  38. Segment(H_{2}, D_{2}) → k_{6}
  39. Intersect(h_{4}, i_{4}) → J_{2}
  40. Intersect(f_{4}, i_{4}) → K_{2}
  41. Intersect(f_{4}, j_{6}) → L_{2}
  42. Intersect(g_{4}, j_{6}) → M_{2}
  43. Intersect(g_{4}, k_{6}) → N_{2}
  44. Intersect(h_{4}, k_{6}) → O_{2}
  45. Polygon(K_{2}, L_{2}, M_{2}, N_{2}, O_{2}, J_{2}) → kuusik_{2}, k_{7}, l_{4}, m_{4}, n_{4}, o_{4}, j_{7}
  46. Polygon(I_{2}, O_{2}, H_{2}, N_{2}, D_{2}, J_{2}, E_{2}, K_{2}, F_{2}, L_{2}, G_{2}, M_{2}) → tahti_{2}, i_{5}, m_{5}, h_{5}, n_{5}, o_{5}, d_{4}, j_{8}, e_{4}, k_{8}, f_{5}, l_{5}, g_{5}
  47. Polygon(A_{4}, B_{4}, C_{4}) → kolmio_{4}, c_{4}, a_{4}, b_{4}
  48. Segment(E_{3}, G_{3}) → f_{6}
  49. Segment(G_{3}, I_{3}) → g_{6}
  50. Segment(I_{3}, E_{3}) → h_{6}
  51. Segment(D_{3}, F_{3}) → i_{6}
  52. Segment(F_{3}, H_{3}) → j_{9}
  53. Segment(H_{3}, D_{3}) → k_{9}
  54. Intersect(h_{6}, i_{6}) → J_{3}
  55. Intersect(f_{6}, i_{6}) → K_{3}
  56. Intersect(f_{6}, j_{9}) → L_{3}
  57. Intersect(g_{6}, j_{9}) → M_{3}
  58. Intersect(g_{6}, k_{9}) → N_{3}
  59. Intersect(h_{6}, k_{9}) → O_{3}
  60. Polygon(K_{3}, L_{3}, M_{3}, N_{3}, O_{3}, J_{3}) → kuusik_{3}, k_{10}, l_{6}, m_{6}, n_{6}, o_{6}, j_{10}
  61. Polygon(I_{3}, O_{3}, H_{3}, N_{3}, D_{3}, J_{3}, E_{3}, K_{3}, F_{3}, L_{3}, G_{3}, M_{3}) → tahti_{3}, i_{7}, m_{7}, h_{7}, n_{7}, o_{7}, d_{5}, j_{11}, e_{5}, k_{11}, f_{7}, l_{7}, g_{7}
  62. Polygon(I, N, H) → k1, h_8, i_8, n_8
  63. Rotate(I, 180°) → I'
  64. Rotate(N, 180°) → N'
  65. Rotate(H, 180°) → H'
  66. Polygon(D, E, F, G, H, I) → kuvio1, d_6, e_6, f_8, g_8, h_9, i_9
  67. Polygon(A_{5}, B_{5}, C_{5}) → kolmio_{5}, c_{5}, a_{5}, b_{5}
  68. Segment(E_{4}, G_{4}) → f_{9}
  69. Segment(G_{4}, I_{4}) → g_{9}
  70. Segment(I_{4}, E_{4}) → h_{10}
  71. Segment(D_{4}, F_{4}) → i_{10}
  72. Segment(F_{4}, H_{4}) → j_{12}
  73. Segment(H_{4}, D_{4}) → k_{12}
  74. Intersect(h_{10}, i_{10}) → J_{4}
  75. Intersect(f_{9}, i_{10}) → K_{4}
  76. Intersect(f_{9}, j_{12}) → L_{4}
  77. Intersect(g_{9}, j_{12}) → M_{4}
  78. Intersect(g_{9}, k_{12}) → N_{4}
  79. Intersect(h_{10}, k_{12}) → O_{4}
  80. Polygon(K_{4}, L_{4}, M_{4}, N_{4}, O_{4}, J_{4}) → kuusik_{4}, k_{13}, l_{8}, m_{8}, n_{9}, o_{8}, j_{13}
  81. Polygon(I_{4}, O_{4}, H_{4}, N_{4}, D_{4}, J_{4}, E_{4}, K_{4}, F_{4}, L_{4}, G_{4}, M_{4}) → tahti_{4}, i_{11}, m_{9}, h_{11}, n_{10}, o_{9}, d_{8}, j_{14}, e_{8}, k_{14}, f_{10}, l_{9}, g_{10}
  82. Polygon(I_{4}, N_{4}, H_{4}) → k1_{1}, h_{12}, i_{12}, n_{11}
  83. Polygon(D_{4}, E_{4}, F_{4}, G_{4}, H_{4}, I_{4}) → kuvio1_{1}, d_{9}, e_{9}, f_{11}, g_{11}, h_{13}, i_{13}
  84. Polygon(A_{6}, B_{6}, C_{6}) → kolmio_{6}, c_{6}, a_{6}, b_{6}
  85. Segment(E_{5}, G_{5}) → f_{12}
  86. Segment(G_{5}, I_{5}) → g_{12}
  87. Segment(I_{5}, E_{5}) → h_{14}
  88. Segment(D_{5}, F_{5}) → i_{14}
  89. Segment(F_{5}, H_{5}) → j_{15}
  90. Segment(H_{5}, D_{5}) → k_{15}
  91. Intersect(h_{14}, i_{14}) → J_{5}
  92. Intersect(f_{12}, i_{14}) → K_{5}
  93. Intersect(f_{12}, j_{15}) → L_{5}
  94. Intersect(g_{12}, j_{15}) → M_{5}
  95. Intersect(g_{12}, k_{15}) → N_{5}
  96. Intersect(h_{14}, k_{15}) → O_{5}
  97. Polygon(K_{5}, L_{5}, M_{5}, N_{5}, O_{5}, J_{5}) → kuusik_{5}, k_{16}, l_{10}, m_{10}, n_{12}, o_{10}, j_{16}
  98. Polygon(I_{5}, O_{5}, H_{5}, N_{5}, D_{5}, J_{5}, E_{5}, K_{5}, F_{5}, L_{5}, G_{5}, M_{5}) → tahti_{5}, i_{15}, m_{11}, h_{15}, n_{13}, o_{11}, d_{11}, j_{17}, e_{11}, k_{17}, f_{13}, l_{11}, g_{13}
  99. Polygon(I_{5}, N_{5}, H_{5}) → k1_{2}, h_{16}, i_{16}, n_{14}
  100. Polygon(D_{5}, E_{5}, F_{5}, G_{5}, H_{5}, I_{5}) → kuvio1_{2}, d_{12}, e_{12}, f_{14}, g_{14}, h_{17}, i_{17}
  101. Polygon(A_{7}, B_{7}, C_{7}) → kolmio_{7}, c_{7}, a_{7}, b_{7}
  102. Segment(E_{6}, G_{6}) → f_{15}
  103. Segment(G_{6}, I_{6}) → g_{15}
  104. Segment(I_{6}, E_{6}) → h_{18}
  105. Segment(D_{6}, F_{6}) → i_{18}
  106. Segment(F_{6}, H_{6}) → j_{18}
  107. Segment(H_{6}, D_{6}) → k_{18}
  108. Intersect(h_{18}, i_{18}) → J_{6}
  109. Intersect(f_{15}, i_{18}) → K_{6}
  110. Intersect(f_{15}, j_{18}) → L_{6}
  111. Intersect(g_{15}, j_{18}) → M_{6}
  112. Intersect(g_{15}, k_{18}) → N_{6}
  113. Intersect(h_{18}, k_{18}) → O_{6}
  114. Polygon(K_{6}, L_{6}, M_{6}, N_{6}, O_{6}, J_{6}) → kuusik_{6}, k_{19}, l_{12}, m_{12}, n_{15}, o_{12}, j_{19}
  115. Polygon(I_{6}, O_{6}, H_{6}, N_{6}, D_{6}, J_{6}, E_{6}, K_{6}, F_{6}, L_{6}, G_{6}, M_{6}) → tahti_{6}, i_{19}, m_{13}, h_{19}, n_{16}, o_{13}, d_{13}, j_{20}, e_{13}, k_{20}, f_{16}, l_{13}, g_{16}
  116. Polygon(I_{6}, N_{6}, H_{6}) → k1_{3}, h_{20}, i_{20}, n_{17}
  117. Polygon(D_{6}, E_{6}, F_{6}, G_{6}, H_{6}, I_{6}) → kuvio1_{3}, d_{14}, e_{14}, f_{17}, g_{17}, h_{21}, i_{21}
  118. Line(D, F) → r_1
  119. Line(I, G) → s_1
  120. Point(s_1) → P
  121. Line(E, P) → t_1
  122. Point(t_1) → S
  123. Intersect(r_1, s_1) → U
  124. Polygon(A_{8}, B_{8}, C_{8}) → kolmio_{8}, c_{8}, a_{8}, b_{8}
  125. Segment(E_{7}, G_{7}) → f_{18}
  126. Segment(G_{7}, I_{7}) → g_{18}
  127. Segment(I_{7}, E_{7}) → h_{22}
  128. Segment(D_{7}, F_{7}) → i_{22}
  129. Segment(F_{7}, H_{7}) → j_{21}
  130. Segment(H_{7}, D_{7}) → k_{21}
  131. Intersect(h_{22}, i_{22}) → J_{7}
  132. Intersect(f_{18}, i_{22}) → K_{7}
  133. Intersect(f_{18}, j_{21}) → L_{7}
  134. Intersect(g_{18}, j_{21}) → M_{7}
  135. Intersect(g_{18}, k_{21}) → N_{7}
  136. Intersect(h_{22}, k_{21}) → O_{7}
  137. Polygon(K_{7}, L_{7}, M_{7}, N_{7}, O_{7}, J_{7}) → kuusik_{7}, k_{22}, l_{14}, m_{14}, n_{18}, o_{14}, j_{22}
  138. Polygon(I_{7}, O_{7}, H_{7}, N_{7}, D_{7}, J_{7}, E_{7}, K_{7}, F_{7}, L_{7}, G_{7}, M_{7}) → tahti_{7}, i_{23}, m_{15}, h_{23}, n_{19}, o_{15}, d_{15}, j_{23}, e_{15}, k_{23}, f_{19}, l_{15}, g_{19}
  139. Polygon(I_{7}, N_{7}, H_{7}) → k1_{4}, h_{24}, i_{24}, n_{20}
  140. Polygon(D_{7}, E_{7}, F_{7}, G_{7}, H_{7}, I_{7}) → kuvio1_{4}, d_{16}, e_{16}, f_{20}, g_{20}, h_{25}, i_{25}
  141. Line(D_{7}, F_{7}) → r_{3}
  142. Line(I_{7}, G_{7}) → s_{2}
  143. Line(C_{8}, B_{8}) → p_{4}
  144. Polygon(A_{9}, B_{9}, C_{9}) → kolmio_{9}, c_{9}, a_{9}, b_{9}
  145. Line(C_{9}, B_{9}) → p_{5}
  146. Line(F, H) → p_2
  147. Line(E, G) → q_2
  148. Line(D, H) → r_2
  149. Intersect(t_1, p_2) → Q
  150. Intersect(q_2, r_2) → R
  151. Intersect(m_1, g_1) → T
  152. Polygon(U, Q, R) → k2, r_4, u_8, q_4
  153. Polygon(A_{10}, B_{10}, C_{10}) → kolmio_{10}, c_{10}, a_{10}, b_{10}
  154. Polygon(A_{11}, B_{11}, C_{11}) → kolmio_{11}, c_{11}, a_{11}, b_{11}
  155. Polygon(A_{12}, B_{12}, C_{12}) → kolmio_{12}, c_{12}, a_{12}, b_{12}
  156. Segment(E_{8}, G_{8}) → f_{21}
  157. Segment(G_{8}, I_{8}) → g_{21}
  158. Segment(I_{8}, E_{8}) → h_{26}
  159. Segment(D_{8}, F_{8}) → i_{26}
  160. Segment(F_{8}, H_{8}) → j_{24}
  161. Segment(H_{8}, D_{8}) → k_{24}
  162. Polygon(D_{8}, E_{8}, F_{8}, G_{8}, H_{8}, I_{8}) → kuvio1_{5}, d_{17}, e_{17}, f_{22}, g_{22}, h_{27}, i_{27}
  163. Line(D_{8}, F_{8}) → r_{5}
  164. Line(I_{8}, G_{8}) → s_{3}
  165. Line(F_{8}, H_{8}) → p_{6}
  166. Line(E_{8}, G_{8}) → q_{5}
  167. Line(D_{8}, H_{8}) → r_{6}
  168. Polygon(A_{13}, B_{13}, C_{13}) → kolmio_{13}, c_{13}, a_{13}, b_{13}
  169. Segment(E_{9}, G_{9}) → f_{23}
  170. Segment(G_{9}, I_{9}) → g_{23}
  171. Segment(I_{9}, E_{9}) → h_{28}
  172. Segment(D_{9}, F_{9}) → i_{28}
  173. Segment(F_{9}, H_{9}) → j_{25}
  174. Segment(H_{9}, D_{9}) → k_{25}
  175. Polygon(D_{9}, E_{9}, F_{9}, G_{9}, H_{9}, I_{9}) → kuvio1_{6}, d_{18}, e_{18}, f_{24}, g_{24}, h_{29}, i_{29}
  176. Line(D_{9}, F_{9}) → r_{7}
  177. Line(I_{9}, G_{9}) → s_{5}
  178. Line(F_{9}, H_{9}) → p_{7}
  179. Line(E_{9}, G_{9}) → q_{6}
  180. Line(D_{9}, H_{9}) → r_{8}

AI Drawing Prompt Reference

Draw a hexagon inside a triangle, containing an inscribed hexagon and an inscribed star. The hexagon's side length is the same as the triangle's side length, and its vertices correspond to the triangle's vertices. The star's side length is the same as the hexagon's side length, and its vertices correspond to the hexagon's vertices. Construct the figure in the following order: first draw a triangle, then draw an inscribed hexagon, and finally draw an inscribed star.
The prompt above was compiled from the original drawing commands and is for reference only.