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三角形莱莫恩点与Lemoine双曲线的几何构造

三角形莱莫恩点与Lemoine双曲线的几何构造
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2023-03-20 18:34
测试改名0圆角向量几何作图
所属专题 geometriadinamica.es
本范例演示了三角形内部特殊点——莱莫恩点（Lemoine点，又称重心）的构造方法及其相关几何性质。涉及的核心数学概念包括：三角形的中线、角平分线、相似三角形性质以及莱莫恩点与双曲线的关系。作图要点：使用中点命令(Midpoint)构造三条中线，角平分线命令(AngularBisector)作角平分线，对称线命令(Mirror)构造相似中线(simediana)，结合多边形(Polygon)、圆(Circle)、双曲线(Hyperbola)等GeoGebra命令，展示了莱莫恩点作为三角形重心的特殊性质及其与Kiepert双曲线的关联。适合中学高年级及大学解析几何课程使用，可作为深入理解三角形特殊点与圆锥曲线关系的优质教学范例。

GeoGebra 命令及参数

  1. Midpoint(A, C) → M3
  2. Angle(C, B, A) → γ
  3. Angle(A, C, B) → β
  4. Angle(B, A, C) → α
  5. Midpoint(C, B) → M1
  6. Line(A, M1) → mediana1
  7. AngularBisector(C, A, B) → bisectrizV1
  8. Mirror(mediana1, bisectrizV1) → simediana1
  9. AngularBisector(A, C, B) → bisectrizV2
  10. Midpoint(A, B) → M2
  11. If(δ ≟ α, M1, If[δ ≟ β, M2, M3]) → M
  12. Line(C, M2) → mediana2
  13. Mirror(mediana2, bisectrizV2) → simediana2
  14. Intersect(simediana1, simediana2) → Lemoine
  15. Line(M, Lemoine) → e
  16. Polygon(A, C, B) → isleta, v3, v1, v2
  17. LineBisector(v1) → mediatriz2
  18. LineBisector(v2) → mediatriz1
  19. Intersect(mediatriz2, mediatriz1) → O
  20. OrthogonalLine(C, bisectrizV2) → i
  21. Intersect(bisectrizV1, i) → I1
  22. Intersect(bisectrizV1, bisectrizV2) → I
  23. OrthogonalLine(A, bisectrizV1) → j
  24. Intersect(i, j) → I3
  25. Conic(Lemoine, I, I3, I1, O) → h1
  26. FirstAxisLength(h1) → semieje
  27. Focus(h1) → Foco2, Foco1
  28. Hyperbola(Foco1, Foco2, semieje) → h
  29. Asymptote(h) → asintota2, asintota1
  30. Segment(O, B) → R
  31. OrthogonalLine(I, v1) → ortoI
  32. Intersect(v1, ortoI) → Ibase
  33. Segment(I, Ibase) → r
  34. Center(h1) → X110
  35. OrthogonalLine(G, v3) → g3
  36. Intersect(v3, g3) → G3
  37. Vector(G, G3) → u3
  38. Ray(P, R312) → s312
  39. Intersect(v3, s312) → S312
  40. Circle(P, d) → circuloP
  41. Intersect(circuloP, v3, 1) → P31
  42. Intersect(circuloP, v3, 2) → P32
  43. Midpoint(P31, P32) → PM3
  44. Ray(P, PM3) → sM3
  45. Intersect(circuloP, sM3) → PCM3,
  46. CircumcircleSector(P31, PCM3, P32) → sector2cE
  47. Polygon(P, P31, P32) → tri2c, z2c2, z2c3, z2c1
  48. Area(A, C, B) → areai
  49. Intersect(circuloP, v1, 1) → P11
  50. If(P11 ≟ P11, true, false) → p11
  51. Intersect(circuloP, v1, 2) → P12
  52. If(P12 ≟ P12, true, false) → p12
  53. Midpoint(P11, P12) → PM1
  54. Ray(P, PM1) → sM1
  55. Intersect(circuloP, sM1) → PCM1,
  56. CircumcircleSector(P11, PCM1, P12) → sector2aE
  57. Polygon(P, P11, P12) → tri2a, z2a2, z2a3, z2a1
  58. OrthogonalLine(G, v1) → g1
  59. Intersect(v1, g1) → G1
  60. Vector(G, G1) → u1
  61. Ray(P, R112) → s112
  62. Intersect(circuloP, s112) → P112, Cir1
  63. CircumcircleSector(P11, P112, P12) → sector2a
  64. Intersect(v1, s112) → S112
  65. If(S112 ≟ S112, sector2a - tri2a, sector2a + tri2a) → area2a
  66. Intersect(circuloP, v2, 1) → P22
  67. If(P22 ≟ P22, true, false) → p22
  68. Intersect(circuloP, v2, 2) → P21
  69. If(P21 ≟ P21, true, false) → p21
  70. Midpoint(P21, P22) → PM2
  71. Ray(P, PM2) → sM2
  72. Intersect(circuloP, sM2) → PCM2, Cir2
  73. CircumcircleSector(P21, PCM2, P22) → sector2bE
  74. Polygon(P, P21, P22) → tri2b, z2b2, z2b3, z2b1
  75. OrthogonalLine(G, v2) → g2
  76. Intersect(v2, g2) → G2
  77. Vector(G, G2) → u2
  78. Ray(P, R212) → s212
  79. Intersect(circuloP, s212) → P212, Cir3
  80. CircumcircleSector(P21, P212, P22) → sector2b
  81. Intersect(v2, s212) → S212
  82. If(S212 ≟ S212, sector2b - tri2b, sector2b + tri2b) → area2b
  83. If(P32 ≟ P32, true, false) → p32
  84. If(P31 ≟ P31, true, false) → p31
  85. If(p11 ∧ p12 ∧ p21 ∧ p22 ∧ p31 ∧ p32, true, false) → pos2E
  86. Intersect(circuloP, s312) → P312, Cir4
  87. CircumcircleSector(P31, P312, P32) → sector2c
  88. If(S312 ≟ S312, sector2c - tri2c, sector2c + tri2c) → area2c
  89. If(pos2E, aread - area2aE - area2bE - area2cE, area2) → area2E
  90. Segment(P, A) → dV1
  91. Segment(P, C) → dV2
  92. Ray(P, M3) → fuera3
  93. Segment(P, B) → dV3
  94. If(dV1 ≤ d ∧ dV2 ≤ d ∧ dV3 ≤ d, true, false) → pos5
  95. If(pos5, areai, 0) → area5
  96. If(area5 < areai, area5, areai) → area5T
  97. Intersect(v2, fuera3) → Corte32
  98. Intersect(v1, fuera3) → Corte31
  99. If(p12 ∧ (¬p11), P12, P11) → P1
  100. If(P1 ≟ P1, true, false) → p1
  101. If(Distance[P21, P1] > Distance[P22, P1], P22, P21) → P21'
  102. Polygon(P, P1, P21') → tri621, ztri621c, ztri621b, ztri621a
  103. If(Distance[P21, P1] > Distance[P22, P1], P21, P22) → P22'
  104. If(p22 ∧ (¬p21), P22, P21) → P2
  105. Midpoint(P2, P1) → V3b
  106. Ray(P, V3b) → sV3
  107. Intersect(circuloP, sV3, 1) → V3c
  108. CircumcircleSector(P2, V3c, P1) → sector1c
  109. Polygon(B, P1, P2) → tri1c, z1c2, z1c3, z1c1
  110. Polygon(P, P1, P2) → tri01c, z01c2, z01c3, z01c1
  111. If(Distance[V3b, B] < dV3 ∧ Distance[A, P] < d, aread - sector1c + tri1c + tri01c, tri1c + sector1c - tri01c) → area1c
  112. Polygon(C, A, P2, P1) → cuadri3, zcuadri3a, zcuadri3b, zcuadri3c, zcuadri3d
  113. If(Distance[P11, P2] > Distance[P12, P2], P12, P11) → P11'
  114. Polygon(P, P2, P11') → tri611, ztri611b, ztri611a, ztri611c
  115. If(Distance[P11, P2] > Distance[P12, P2], P11, P12) → P12'
  116. If(P2 ≟ P2, true, false) → p2
  117. If(Distance[P31, P2] > Distance[P32, P2], P32, P31) → P31'
  118. Polygon(P, P2, P31') → tri632, ztri632a, ztri632c, ztri632b
  119. If(Distance[P31, P2] > Distance[P32, P2], P31, P32) → P32'
  120. Polygon(B, P1, P32', P31', P2) → penta3, zpenta3e, zpenta3d, zpenta3c, zpenta3b, zpenta3a
  121. Polygon(P, P1, P32') → tri631, ztri631b, ztri631c, ztri631a
  122. If(p31 ∧ p32 ∧ p1 ∧ p2 ∧ ((¬p11) ∨ (¬p12)) ∧ ((¬p21) ∨ (¬p22)), true, false) → pos6c
  123. If(p11 ∧ p12 ∧ p21 ∧ p22 ∨ p11 ∧ p12 ∧ p31 ∧ p32 ∨ p21 ∧ p22 ∧ p31 ∧ p32, true, false) → pos2
  124. If(p32 ∧ (¬p31), P32, P31) → P3
  125. Polygon(C, P1, P21', P22', P3) → penta2, zpenta2c, zpenta2b, zpenta2a, zpenta2e, zpenta2d
  126. Polygon(P, P3, P22') → tri622, ztri622c, ztri622a, ztri622b
  127. Midpoint(P1, P3) → V2b
  128. Ray(P, V2b) → sV2
  129. Intersect(circuloP, sV2, 1) → V2c
  130. CircumcircleSector(P1, V2c, P3) → sector1b
  131. Polygon(C, P1, P3) → tri1b, z1b2, z1b3, z1b1
  132. Polygon(P, P1, P3) → tri01b, z01b2, z01b3, z01b1
  133. If(Distance[V2b, C] < dV2 ∧ Distance[A, P] < d, aread - sector1b + tri1b + tri01b, tri1b + sector1b - tri01b) → area1b
  134. Polygon(A, B, P1, P3) → cuadri2, zcuadri2a, zcuadri2b, zcuadri2c, zcuadri2d
  135. Polygon(A, P2, P11', P12', P3) → penta1, zpenta1a, zpenta1b, zpenta1c, zpenta1d, zpenta1e
  136. Polygon(P, P3, P12') → tri612, ztri612c, ztri612b, ztri612a
  137. Midpoint(P2, P3) → V1b
  138. Ray(P, V1b) → sV1
  139. Intersect(circuloP, sV1, 1) → V1c
  140. CircumcircleSector(P2, V1c, P3) → sector1a
  141. Polygon(P, P2, P3) → tri01a, z01a2, z01a3, z01a1
  142. Polygon(A, P2, P3) → tri1a, z1a2, tri1a3, z1a1
  143. If(Distance[V1b, A] < dV1 ∧ Distance[B, P] < d, aread - sector1a + tri1a + tri01a, tri1a + sector1a - tri01a) → area1a
  144. If(area1a ≟ area1a, area1a, If[area1b ≟ area1b, area1b, area1c]) → area1
  145. If(area1 < areai, area1, areai) → area1T
  146. Polygon(C, B, P2, P3) → cuadri1, zcuadri1a, zcuadri1b, zcuadri1c, zcuadri1d
  147. If(area4 < areai, area4, areai) → area4T
  148. If(P3 ≟ P3, true, false) → p3
  149. If(p21 ∧ p22 ∧ p1 ∧ p3 ∧ ((¬p11) ∨ (¬p12)) ∧ ((¬p31) ∨ (¬p32)), true, false) → pos6b
  150. If((¬p3) ∧ p1 ∧ ((¬p11) ∨ (¬p12)) ∧ p2 ∧ ((¬p21) ∨ (¬p22)), true, false) → pos1c
  151. If((¬p2) ∧ p1 ∧ ((¬p11) ∨ (¬p12)) ∧ p3 ∧ ((¬p31) ∨ (¬p32)), true, false) → pos1b
  152. If((¬p1) ∧ p2 ∧ ((¬p21) ∨ (¬p22)) ∧ p3 ∧ ((¬p31) ∨ (¬p32)), true, false) → pos1a
  153. If(pos1a ∨ pos1b ∨ pos1c, true, false) → pos1
  154. If(pos1 ∧ dV2 ≤ d ∧ dV3 ≤ d, true, false) → pos4c
  155. If(pos1 ∧ dV1 ≤ d ∧ dV3 ≤ d, true, false) → pos4b
  156. If(pos1 ∧ dV1 ≤ d ∧ dV2 ≤ d, true, false) → pos4a
  157. If(pos4a ∨ pos4b ∨ pos4c, true, false) → pos4
  158. If(p1 ∧ (¬p2) ∧ (¬p3) ∨ (¬p1) ∧ p2 ∧ (¬p3) ∨ (¬p1) ∧ (¬p2) ∧ p3, true, false) → pos3
  159. If(p11 ∧ p12 ∧ p2 ∧ p3 ∧ ((¬p21) ∨ (¬p22)) ∧ ((¬p31) ∨ (¬p32)), true, false) → pos6a
  160. If(pos6a ∨ pos6b ∨ pos6c, true, false) → pos6
  161. CircumcircleSector(P2, P312, P1) → sector6c
  162. If(pos6c, penta3 + sector6c - sector2c - tri631 - tri632, 0) → area6c
  163. CircumcircleSector(P1, P212, P3) → sector6b
  164. If(pos6b, penta2 + sector6b - sector2b - tri621 - tri622, 0) → area6b
  165. CircumcircleSector(P2, P112, P3) → sector6a
  166. If(pos6a, penta1 + sector6a - sector2a - tri611 - tri612, 0) → area6a
  167. If(area2E < areai, area2E, areai) → area2T
  168. If(area6 < areai, area6, areai) → area6T
  169. Ray(P, M2) → fuera2
  170. Intersect(v1, fuera2) → Corte21
  171. Intersect(v3, fuera2) → Corte23
  172. Ray(P, M1) → fuera1
  173. Intersect(v3, fuera1) → Corte13
  174. Intersect(v2, fuera1) → Corte12
  175. Line(B, Lemoine) → simediana3
  176. If(sector2c > aread / 2, sector2c + tri2c, sector2c) → sector2C
  177. If(areaT > 0 ∨ Corte12 ≟ Corte12 ∨ Corte13 ≟ Corte13 ∨ Corte21 ≟ Corte21 ∨ Corte23 ≟ Corte23 ∨ Corte31 ≟ Corte31 ∨ Corte32 ≟ Corte32, areaT, aread) → area
  178. Angle(isleta) → ε, ζ, η
  179. Line(B, M3) → mediana3
  180. Intersect(bisectrizV2, j) → I2
  181. Circle(O, R) → circunscrita
  182. Line(G, O) → euler
  183. Mirror(euler, v2) → eulerAB
  184. Mirror(euler, v1) → eulerBC
  185. Intersect(h1, e, 2) → H
  186. Intersect(h1, euler, 1) → J
  187. Segment(J, I) → a
  188. Intersect(euler, eulerBC) → EE
  189. Segment(O, B) → b
  190. Circle(I, Ibase) → f
  191. Line(I, B) → c
  192. OrthogonalLine(B, c) → g
  193. Point(asintota2) → D
  194. Point(asintota1) → E
  195. Angle(D, X110, E) → θ
  196. Mirror(euler, v3) → eulerAC

AI 绘图提示语参考

在给定三角形中，先作三条角的角平分线与中线：取两边对应的角平分方向，与从顶点到对边中点的连线共同建立对称线。用镜像关系得到两条与角平分线相关的对称线，它们的交点即为 Lemoine 点。再取该中心点与由角平分线及其正交线交出的若干辅点，构造一条经过这些关键点的二次曲线，使其中心与焦点可由构造得到，并与另一条双曲线相关联：画出该双曲线及其两条渐近线、焦点位置。最后在三角形的几何区域中标出相关线段与圆弧扇形，整体呈现 Lemoine 点—Lemoine 双曲线的几何联系。
以上提示语为根据原始绘图命令整理而来，仅供参考。