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Explore Geometric Theorems with Dragging Points and Slider

Explore Geometric Theorems with Dragging Points and Slider
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2015-02-22 02:15
TriangleCircleTrigonometryGeometric Construction
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This example demonstrates geometric properties of triangles and incircles by dragging points and a slider. It involves core concepts like triangles, incircles, geometric constructions, and trigonometry. Constructed using commands like Point, CircleArc, Incircle, Midpoint, Segment, OrthogonalLine, Angle, Rotate. Suitable for high school and above students to explore geometric theorems and deepen understanding of geometric constructions.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. CircleArc(B, D, A) → c
  4. Point(c) → E
  5. Polygon(A, E, 4) → poly1, a, b, d, e, F, G
  6. Polygon(E, C, 4) → poly2, f, g, h, i, H, I
  7. Polygon(C, A, 4) → poly3, j, k, l, m, J, K
  8. Incircle(A, E, C) → p
  9. Intersect(p, f) → L, P
  10. Intersect(p, a) → M,
  11. Intersect(p, j) → N, O
  12. Midpoint(F, A) → Q
  13. Midpoint(E, H) → R
  14. Mirror(M, Q) → M'
  15. Mirror(L, R) → L'
  16. Segment(L, L') → n
  17. OrthogonalLine(R, n) → q
  18. Segment(M', M) → r
  19. OrthogonalLine(Q, r) → s
  20. Intersect(s, e) → S
  21. Intersect(s, b) → T
  22. Intersect(q, i) → U
  23. Intersect(q, g) → V
  24. Polygon(P, V, R) → poly4, r_1, p_1, v
  25. Polygon(P, U, R) → poly5, r_2, p_2, u
  26. Polygon(U, R, L') → poly6, l', u_1, r_3
  27. Polygon(R, V, L') → poly7, l'_1, r_4, v_1
  28. Polygon(U, I, L') → poly8, l'_2, u_2, i_1
  29. Polygon(U, E, P) → poly9, p_3, u_3, e_1
  30. Polygon(L', H, V) → poly10, v_2, l'_3, h_1
  31. Polygon(C, V, P) → poly11, p_4, c_1, v_3
  32. Polygon(S, M, T, M') → poly12, s_1, m_1, t, m'
  33. Polygon(S, A, M) → poly13, m_2, s_2, a_1
  34. Polygon(T, E, M) → poly14, m_3, t_1, e_2
  35. Polygon(T, M', F) → poly15, f_1, t_2, m'_1
  36. Polygon(G, M', S) → poly16, s_3, g_1, m'_2
  37. Angle(N, A, M) → α
  38. Angle(P, C, N) → β
  39. If(0 ≤ o ≤ 1, 1 - o, If[o > 1, 0, 1]) → d1
  40. Dilate(M, d1, K) → M'_1
  41. Dilate(S, d1, J) → S'
  42. Dilate(M', d1, A) → M''
  43. Dilate(T, d1, C) → T'
  44. Vector(M, M'_1) → w
  45. Vector(S, S') → w_1
  46. Vector(M', M'') → b_1
  47. Vector(T, T') → d_1
  48. Translate(S, w) → S'_1
  49. Translate(A, w) → A'
  50. Translate(M, w) → M'_2
  51. Polygon(S'_1, A', M'_2) → poly13', m'_3, s'_1, a'
  52. If(0 ≤ o ≤ 1, o - 0, If[o > 1, 1, 0]) → r1
  53. Rotate(S'_1, (-((α * r1))), M'_1) → W
  54. Rotate(A', (-((α * r1))), M'_1) → A''
  55. Rotate(M'_2, (-((α * r1))), M'_1) → Z
  56. Polygon(W, A'', Z) → poly13'', z_1, w_2, a''
  57. Translate(G, w_1) → G'
  58. Translate(M', w_1) → M''_1
  59. Translate(S, w_1) → S'_2
  60. Polygon(G', M''_1, S'_2) → poly16', s'_2, g', m''_1
  61. Translate(T, b_1) → T'_1
  62. Translate(M', b_1) → M''_2
  63. Translate(F, b_1) → F'
  64. Polygon(T'_1, M''_2, F') → poly15', f', t'_1, m''_2
  65. Translate(T, d_1) → T'_2
  66. Translate(E, d_1) → E'
  67. Translate(M, d_1) → M'_3
  68. Polygon(T'_2, E', M'_3) → poly14', m'_4, t'_2, e'
  69. Rotate(G', (-((r1 * α))), S') → G''
  70. Rotate(M''_1, (-((r1 * α))), S') → A_1
  71. Rotate(S'_2, (-((r1 * α))), S') → B_1
  72. Polygon(G'', A_1, B_1) → poly16'', b_2, g'', a_2
  73. Rotate(T'_1, (-((r1 * α))), M'') → C_1
  74. Rotate(M''_2, (-((r1 * α))), M'') → D_1
  75. Rotate(F', (-((r1 * α))), M'') → F''
  76. Polygon(C_1, D_1, F'') → poly15'', f'', c_2, d_2
  77. Rotate(T'_2, (-((r1 * α))), T') → E_1
  78. Rotate(E', (-((r1 * α))), T') → E''
  79. Rotate(M'_3, (-((r1 * α))), T') → F_1
  80. Polygon(E_1, E'', F_1) → poly14'', f_2, e_3, e''
  81. If(1 ≤ o ≤ 2, 2 - o, If[o > 2, 0, 1]) → d2
  82. Dilate(E, d2, G'') → E'_1
  83. Dilate(C, d2, A'') → C'
  84. Dilate(H, d2, E'') → H'
  85. Dilate(I, d2, N) → I'
  86. Vector(E, E'_1) → j_1
  87. Vector(I, I') → k_1
  88. Vector(H, H') → l_1
  89. Vector(C, C') → n_1
  90. Translate(U, j_1) → U'
  91. Translate(E, j_1) → E'_2
  92. Translate(P, j_1) → P'
  93. Polygon(U', E'_2, P') → poly9', p', u', e'_2
  94. If(1 ≤ o ≤ 2, o - 1, If[o > 2, 1, 0]) → r2
  95. Rotate(U', (r2 * β), E'_1) → U''
  96. Rotate(E'_2, (r2 * β), E'_1) → G_1
  97. Rotate(P', (r2 * β), E'_1) → P''
  98. Polygon(U'', G_1, P'') → poly9'', p'', u'', g_2
  99. Translate(U, k_1) → U'_1
  100. Translate(I, k_1) → I'_1
  101. Translate(L', k_1) → L''
  102. Polygon(U'_1, I'_1, L'') → poly8', l'', u'_1, i'_1
  103. Rotate(U'_1, (r2 * β), I') → H_1
  104. Rotate(I'_1, (r2 * β), I') → I_1
  105. Rotate(L'', (r2 * β), I') → L'''
  106. Polygon(H_1, I_1, L''') → poly8'', l''', h_2, i_2
  107. Translate(L', l_1) → L''_1
  108. Translate(H, l_1) → H'_1
  109. Translate(V, l_1) → V'
  110. Polygon(L''_1, H'_1, V') → poly10', v', l''_1, h'_1
  111. Rotate(L''_1, (r2 * β), H') → J_1
  112. Rotate(H'_1, (r2 * β), H') → K_1
  113. Rotate(V', (r2 * β), H') → V''
  114. Polygon(J_1, K_1, V'') → poly10'', v'', j_2, k_2
  115. Translate(C, n_1) → C'_1
  116. Translate(V, n_1) → V'_1
  117. Translate(P, n_1) → P'_1
  118. Polygon(C'_1, V'_1, P'_1) → poly11', p'_1, c'_1, v'_1
  119. Rotate(C'_1, (r2 * β), C') → L_1
  120. Rotate(V'_1, (r2 * β), C') → M_1
  121. Rotate(P'_1, (r2 * β), C') → N_1
  122. Polygon(L_1, M_1, N_1) → poly11'', n_2, l_2, m_4
  123. If(2 ≤ o ≤ 3, 3 - o, If[o > 3, 0, 1]) → d3
  124. Dilate(P, d3, A_1) → P'_2
  125. Dilate(V, d3, W) → V'_2
  126. Vector(P, P'_2) → q_1
  127. Translate(P, q_1) → P'_3
  128. Translate(U, q_1) → U'_2
  129. Translate(R, q_1) → R'
  130. Polygon(P'_3, U'_2, R') → poly5', r', p'_3, u'_2
  131. If(2 ≤ o ≤ 3, o - 2, If[o > 3, 1, 0]) → r3
  132. Rotate(P'_3, (r3 * β), P'_2) → O_1
  133. Rotate(U'_2, (r3 * β), P'_2) → P_1
  134. Rotate(R', (r3 * β), P'_2) → R''
  135. Polygon(O_1, P_1, R'') → poly5'', r'', o_1, p_5
  136. Vector(V, V'_2) → q_2
  137. Translate(P, q_2) → P'_4
  138. Translate(V, q_2) → V'_3
  139. Translate(R, q_2) → R'_1
  140. Polygon(P'_4, V'_3, R'_1) → poly4', r'_1, p'_4, v'_3
  141. Rotate(P'_4, (r3 * β), V'_2) → Q_1
  142. Rotate(V'_3, (r3 * β), V'_2) → R_1
  143. Rotate(R'_1, (r3 * β), V'_2) → S_1
  144. Polygon(Q_1, R_1, S_1) → poly4'', s_4, q_3, r_5
  145. Dilate(L', d3, F_1) → L''_2
  146. Vector(L', L''_2) → w_3
  147. Translate(R, w_3) → R'_2
  148. Translate(V, w_3) → V'_4
  149. Translate(L', w_3) → L''_3
  150. Polygon(R'_2, V'_4, L''_3) → poly7', l''_3, r'_2, v'_4
  151. Rotate(R'_2, (r3 * β), L''_2) → T_1
  152. Rotate(V'_4, (r3 * β), L''_2) → U_1
  153. Rotate(L''_3, (r3 * β), L''_2) → V_1
  154. Polygon(T_1, U_1, V_1) → poly7'', v_4, t_3, u_4
  155. Dilate(U, d3, C_1) → U'_3
  156. Vector(U, U'_3) → a_3
  157. Translate(U, a_3) → U'_4
  158. Translate(R, a_3) → R'_3
  159. Translate(L', a_3) → L''_4
  160. Polygon(U'_4, R'_3, L''_4) → poly6', l''_4, u'_4, r'_3
  161. Rotate(U'_4, (r3 * β), U'_3) → W_1
  162. Rotate(R'_3, (r3 * β), U'_3) → Z_1
  163. Rotate(L''_4, (r3 * β), U'_3) → A_2
  164. Polygon(W_1, Z_1, A_2) → poly6'', a_4, w_4, z_2
  165. If(3 ≤ o ≤ 4, 4 - o, If[o > 4, 0, 1]) → d4
  166. If(3 ≤ o ≤ 4, o - 3, If[o > 4, 1, 0]) → r4
  167. Dilate(S, d4, T_1) → S'_3
  168. Vector(S, S'_3) → b_3
  169. Translate(S, b_3) → S'_4
  170. Translate(M, b_3) → M'_4
  171. Translate(T, b_3) → T'_3
  172. Translate(M', b_3) → M''_3
  173. Polygon(S'_4, M'_4, T'_3, M''_3) → poly12', s'_4, m'_5, t'_3, m''_3
  174. Rotate(S'_4, (r4 * β), S'_3) → B_2
  175. Rotate(M'_4, (r4 * β), S'_3) → C_2
  176. Rotate(T'_3, (r4 * β), S'_3) → D_2
  177. Rotate(M''_3, (r4 * β), S'_3) → E_2
  178. Polygon(B_2, C_2, D_2, E_2) → poly12'', b_4, c_3, d_3, e_4

AI Drawing Prompt Reference

Draw a triangle ABC and explore the properties of its incircle by dragging points and using a slider. Start by creating point A on the x-axis, then point B at the origin, and draw an arc centered at B intersecting the x-axis at D. Create point E and use points E, C, and A to form a polygon. Construct the incircle p of triangle ABC and find its intersections with the polygon's sides at points L, M, N, and O. Use midpoint and reflection operations to create a series of polygons and segments, forming a complex geometric figure to explore the relationship between the triangle and its incircle.
The prompt above was compiled from the original drawing commands and is for reference only.