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幸运莱昂垂心：三角形垂心与对称性的动态探究

幸运莱昂垂心：三角形垂心与对称性的动态探究
几何画板及绘图内容加载中，请稍等……
2017-02-14 06:19
测试改名0解析几何矩阵变换几何作图
① 本范例通过GeoGebra动态演示三角形三条高线交于垂心（Orthocenter）的经典几何现象，并利用矩阵与对称变换展示垂心与重心、边线之间的位置关系。② 核心数学概念包括三角形的垂心、重心（质心）、垂线、对称性以及泊松分布（作为背景统计标签），涉及直线方程与矩阵变换。③ 作图要点：使用Polygon构造三角形ABC，Distance测量边长，OrthogonalLine构造各边的高线，PolyLine连接相关点；对象包括顶点A、B、C，三角形poly1，边a、b、c，高线f、g、h，垂心D，以及文本text1-text5和列表list1-list3，用于标注和动态显示。④ 适合中学几何教学或大学初等几何课程，用于直观理解垂心性质、对称变换与矩阵在几何中的应用，也可作为交互式探究素材。

GeoGebra 命令及参数

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Distance(B, C) → distanceBC
  3. OrthogonalLine(A, a) → f
  4. OrthogonalLine(B, b) → g
  5. OrthogonalLine(C, c) → h
  6. PolyLine((11.73520089766761, 1.9782675475147982), (NaN, NaN), (14.005414635870013, 1.9782675475147982), (14.264867634521716, 1.9782675475147982), (14.524320633173419, 1.9782675475147982), (14.524320633173419, 1.718814548863095), (14.524320633173419, 1.459361550211392), (14.524320633173419, 1.1999085515596888), (14.524320633173419, 0.9404555529079858), (14.524320633173419, 0.6810025542562826), (14.524320633173419, 0.42154955560457946), (14.524320633173419, 0.16209655695287634), (11.73520089766761, 1.9782675475147982), (14.524320633173419, -0.09735644169882679), true, (11.994653896319313, 1.9782675475147982), (12.318970144633942, 1.9782675475147982), (12.643286392948571, 1.9782675475147982), (12.902739391600274, 1.9782675475147982), (13.162192390251978, 1.9782675475147982), (13.42164538890368, 1.9782675475147982), (13.681098387555384, 1.9782675475147982)) → stroke1
  7. PolyLine((17.248577119016304, 17.091404718976506), (NaN, NaN), (18.610705361937743, 15.79413972571799), (18.805295110926522, 15.859002975380916), (18.935021610252374, 16.053592724369693), (19.064748109578225, 16.24818247335847), (19.259337858567, 16.313045723021396), (19.45392760755578, 16.442772222347248), (19.648517356544556, 16.507635472010172), true, (17.248577119016304, 17.091404718976506), (17.44316686800508, 16.961678219650654), (17.57289336733093, 16.767088470661875), (17.76748311631971, 16.5724987216731), (17.962072865308485, 16.442772222347248), (18.091799364634337, 16.24818247335847), (18.286389113623116, 16.118455974032617), (18.48097886261189, 15.988729474706767)) → stroke2
  8. PolyLine((8.751491413173024, 13.718515736504365), (NaN, NaN), (9.983893156768614, 15.275233728414584), (10.113619656094466, 15.469823477403361), (10.243346155420317, 15.664413226392139), (10.113619656094466, 15.859002975380916), (9.919029907105688, 16.053592724369693), (9.724440158116911, 16.183319223695545), (9.529850409128134, 16.24818247335847), (9.335260660139356, 16.37790897268432), (9.140670911150579, 16.442772222347248), (8.946081162161802, 16.5724987216731), (8.751491413173024, 13.718515736504365), (8.81635466283595, 16.767088470661875), (8.621764913847173, 16.831951720324803), (8.427175164858395, 16.961678219650654), (8.297448665532544, 17.15626796863943), true, (8.881217912498876, 13.913105485493142), (9.075807661487653, 14.172558484144846), (9.27039741047643, 14.367148233133623), (9.400123909802282, 14.5617379821224), (9.529850409128134, 14.756327731111178), (9.659576908453985, 14.950917480099955), (9.854166657442763, 15.080643979425806)) → stroke3
  9. PolyLine((13.032465890926126, 10.994259250661482), (NaN, NaN), (14.394594133847567, 9.437541258751263), (14.524320633173419, 9.242951509762486), (14.718910382162196, 9.048361760773709), (14.9783633808139, 8.853772011784931), (15.172953129802677, 8.659182262796154), (15.367542878791454, 8.464592513807377), (15.562132627780231, 8.2700027648186), (15.756722376769009, 8.140276265492748), (15.951312125757786, 7.9456865165039705), (16.21076512440949, 7.686233517852267), (13.032465890926126, 10.994259250661482), (16.405354873398267, 7.556507018526416), (NaN, NaN), (16.405354873398267, 7.556507018526416), (NaN, NaN), (16.405354873398267, 7.556507018526416), (16.08103862508364, 7.556507018526416), (15.821585626431935, 7.556507018526416), (15.562132627780231, 7.556507018526416), (15.302679629128528, 7.556507018526416), (15.043226630476825, 7.556507018526416), (13.162192390251978, 10.799669501672705), (14.783773631825122, 7.556507018526416), (NaN, NaN), (16.470218123061194, 7.556507018526416), (NaN, NaN), (16.470218123061194, 7.556507018526416), (16.470218123061194, 7.815960017178119), (16.535081372724118, 8.075413015829822), (16.535081372724118, 8.334866014481525), (16.599944622387046, 8.529455763470303), (16.599944622387046, 8.788908762122006), (13.356782139240755, 10.605079752683928), (16.599944622387046, 9.048361760773709), true, (13.551371888229532, 10.41049000369515), (13.681098387555384, 10.215900254706373), (13.810824886881235, 10.021310505717596), (14.005414635870013, 9.761857507065892), (14.20000438485879, 9.63213100774004)) → stroke4