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Carnot Action! = GoGeometry Action 27!

Carnot Action! = GoGeometry Action 27!
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2017-04-17 23:51
TriangleCircleTransformationGeometric ConstructionSlider Animation
① This interactive GeoGebra example dynamically illustrates Carnot's Theorem for the case of an obtuse triangle, allowing users to drag vertices and observe the invariant relationships. ② Core mathematical concepts include circumcircle, incircle, perpendicular distances from vertices to opposite sides, and the algebraic identity relating these distances to the triangle's side lengths and circumradius. ③ The construction uses GeoGebra commands such as Rotate, Segment, Ray, Point, Intersect, LineBisector, OrthogonalLine, Circle, CircleSector, Dilate, CircleArc, Vector, Translate, and Line to build the triangle, auxiliary circles, and perpendicular lines, with dynamic sliders or draggable points for exploration. ④ Suitable for high school and above, or geometry enthusiasts, this example serves as a visual proof and exploration tool for understanding Carnot's theorem, triangle geometry, and the properties of circumscribed and inscribed circles.

GeoGebra Commands & Parameters

  1. Rotate(A, a°, B) → A'
  2. Angle(A, B, A') → α
  3. Segment(B, A) → f
  4. Ray(B, A') → g
  5. Point(g) → C
  6. Segment(A, C) → h
  7. Segment(B, C) → i
  8. Rotate(A, α / 2, B) → A'_1
  9. Angle(A, B, A'_1) → β
  10. Ray(B, A'_1) → j
  11. Angle(C, A, B) → γ
  12. Rotate(C, γ / 2, A) → C'
  13. Angle(C, A, C') → δ
  14. Ray(A, C') → k
  15. Intersect(k, j) → D
  16. LineBisector(h) → l
  17. LineBisector(i) → m
  18. Intersect(l, m) → E
  19. OrthogonalLine(D, i) → n
  20. OrthogonalLine(D, h) → p
  21. Intersect(n, i) → F
  22. Intersect(p, h) → G
  23. Angle(D, G, A) → ε
  24. Angle(D, F, C) → ζ
  25. Min(f, i) → b
  26. Circle(A, (0.2 * b)) → c
  27. Circle(B, (0.2 * b)) → d
  28. Intersect(c, h, 1) → H
  29. Intersect(c, k, 1) → I
  30. Intersect(d, f, 1) → J
  31. Intersect(d, j, 1) → K
  32. CircleSector(A, H, I) → e
  33. CircleSector(B, J, K) → q
  34. Intersect(i, k) → L
  35. Intersect(h, j) → M
  36. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], L) → A'_2
  37. Segment(A, A'_2) → r
  38. Dilate(B, If[1 < o ≤ 2, 2 - o, 0], D) → B'
  39. Segment(B, B') → s
  40. Dilate(D, If[2 < o ≤ 3, 3 - o, 0], M) → D'
  41. Segment(D, D') → t
  42. Rotate(e, If[3 < o ≤ 4, ((o - 3) * γ / 2), γ / 2], A) → e'
  43. Rotate(q, If[4 < o ≤ 5, ((o - 4) * α / 2), α / 2], B) → q'
  44. Circle(D, F) → c_1
  45. Rotate(F, If[5 < o ≤ 7, ((o - 5) * π), π], D) → F'
  46. CircleArc(D, F, F') → d_1
  47. Dilate(E, If[8 < o ≤ 9, 9 - o, 0], C) → E'
  48. Segment(E, E') → f_1
  49. Circle(E, C) → e_1
  50. Rotate(C, If[9 < o ≤ 11, ((o - 9) * π), π], E) → C'_1
  51. CircleArc(E, C, C'_1) → g_1
  52. Min({y(E), y(A), y(B), y(C)}) → u
  53. OrthogonalLine(E, h_1) → j_1
  54. Intersect(h_1, j_1) → N
  55. Intersect(i_1, j_1) → O
  56. OrthogonalLine(E, h) → k_1
  57. OrthogonalLine(E, i) → l_1
  58. OrthogonalLine(E, f) → m_1
  59. Intersect(l_1, i) → P
  60. Intersect(k_1, h) → Q
  61. Intersect(m_1, f) → R
  62. Dilate(E, If[11 < o ≤ 12, 12 - o, 0], P) → E'_1
  63. Dilate(E, If[11 < o ≤ 12, 12 - o, 0], Q) → E'_2
  64. Dilate(E, If[11 < o ≤ 12, 12 - o, 0], R) → E'_3
  65. Segment(E, E'_1) → n_1
  66. Segment(E, E'_2) → p_1
  67. Segment(E, E'_3) → q_1
  68. Angle(E, Q, C) → η
  69. Angle(E, P, C) → θ
  70. Angle(E, R, B) → ι
  71. Dilate(E, If[12 < o ≤ 13, 13 - o, 0], N) → E'_4
  72. Vector(E, E'_4) → v
  73. Translate(E, v) → E'_5
  74. Translate(E', v) → E''
  75. Segment(E'_5, E'') → r_1
  76. Line(E, h_1) → s_1
  77. Intersect(e_1, s_1, 2) → T
  78. Angle(T, E, C) → κ
  79. Rotate(E'_5, If[13 < o ≤ 14, ((-(o - 13)) * κ), (-κ)], N) → U
  80. Rotate(E'', If[13 < o ≤ 14, ((-(o - 13)) * κ), (-κ)], N) → E'''
  81. Segment(U, E''') → t_1
  82. Dilate(D, If[5 < o ≤ 6, 6 - o, 0], F) → D'_1
  83. Segment(D, D'_1) → a_1
  84. Dilate(D, If[5 < o ≤ 6, 6 - o, 0], G) → D'_2
  85. Segment(D, D'_2) → b_1
  86. Dilate(D, If[14 < o ≤ 15, 15 - o, 0], E''') → D'_3
  87. Vector(D, D'_3) → w
  88. Translate(D, w) → D'_4
  89. Translate(D'_1, w) → S
  90. Segment(D'_4, S) → f_2
  91. Line(D, i_1) → g_2
  92. Intersect(c_1, g_2, 2) → V
  93. Angle(V, D, F) → λ
  94. Rotate(D'_4, If[15 < o ≤ 16, ((-(o - 15)) * λ), (-λ)], E''') → W
  95. Rotate(S, If[15 < o ≤ 16, ((-(o - 15)) * λ), (-λ)], E''') → S'
  96. Segment(W, S') → h_2
  97. Circle(O, t_1 + h_2) → c_2
  98. Intersect(c_2, i_1, 2) → Z
  99. Circle(O, 1) → d_2
  100. Circle(Z, 1) → e_2
  101. Intersect(d_2, i_1, 1) → A_1
  102. Intersect(e_2, i_1, 2) → B_1
  103. Ray(O, A_1) → i_2
  104. Ray(Z, B_1) → j_2
  105. Circle(O, q_1) → k_2
  106. Intersect(k_2, i_1, 2) → C_1
  107. Circle(C_1, n_1) → p_2
  108. Intersect(i_1, j_2) → D_1
  109. Intersect(i_1, j_2) → E_1
  110. Intersect(p_2, i_1) → F_1, G_1
  111. Ray(O, A_1) → l_2
  112. Segment(N, S') → m_2
  113. Dilate(N, If[16 < o ≤ 17, 17 - o, 0], O) → N'
  114. Vector(N, N') → u_1
  115. Translate(N, u_1) → N'_1
  116. Translate(S', u_1) → S''
  117. Segment(N'_1, S'') → n_2
  118. Segment(N, N'_1) → q_2
  119. Segment(O, C_1) → r_2
  120. Segment(C_1, Z) → s_2
  121. Circle(G_1, 1) → t_2
  122. Intersect(t_2, j_2, 2) → H_1
  123. Ray(G_1, H_1) → a_2
  124. Segment(Z, G_1) → b_2
  125. Dilate(O, If[17 < o ≤ 18, 18 - o, 0], Z) → O'
  126. Vector(O, O') → v_1
  127. Translate(N, v_1) → N'_2
  128. Translate(N'_1, v_1) → I_1
  129. Segment(N'_2, I_1) → f_3
  130. Dilate(R, If[18 < o ≤ 19, 19 - o, 0], O) → R'
  131. Vector(R, R') → w_1
  132. Translate(E, w_1) → E'_6
  133. Translate(E'_3, w_1) → J_1
  134. Segment(E'_6, J_1) → g_3
  135. Line(R, h_1) → h_3
  136. Intersect(e_1, h_3, 2) → K_1
  137. Angle(E, R, K_1) → μ
  138. Rotate(E'_6, If[19 < o ≤ 20, ((o - 19) * μ), μ], O) → L_1
  139. Rotate(J_1, If[19 < o ≤ 20, ((o - 19) * μ), μ], O) → M_1
  140. Segment(L_1, M_1) → i_3
  141. Dilate(E, If[20 < o ≤ 21, 21 - o, 0], C_1) → E'_7
  142. Vector(E, E'_7) → u_2
  143. Translate(E, u_2) → E'_8
  144. Translate(E'_1, u_2) → N_1
  145. Segment(E'_8, N_1) → j_3
  146. Angle(T, E, P) → τ
  147. Rotate(E'_8, If[21 < o ≤ 22, ((-(o - 21)) * τ), (-τ)], C_1) → O_1
  148. Rotate(N_1, If[21 < o ≤ 22, ((-(o - 21)) * τ), (-τ)], C_1) → P_1
  149. Segment(O_1, P_1) → k_3
  150. Intersect(e_1, s_1, 1) → Q_1
  151. Segment(Z, G_1) → m_3
  152. Segment(C_1, Z) → n_3
  153. Dilate(E, If[22 < o ≤ 23, 23 - o, 0], G_1) → E'_9
  154. Vector(E, E'_9) → v_2
  155. Translate(E, v_2) → E'_{10}
  156. Translate(E'_2, v_2) → R_1
  157. Segment(E'_{10}, R_1) → p_3
  158. Angle(Q, E, Q_1) → χ
  159. Rotate(E'_{10}, If[23 < o ≤ 24, ((o - 23) * χ), χ], G_1) → S_1
  160. Rotate(R_1, If[23 < o ≤ 24, ((o - 23) * χ), χ], G_1) → T_1
  161. Segment(S_1, T_1) → q_3
  162. Polygon(A, B, C) → poly1, c_3, a_3, b_3