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POW: Lucky Leon Orthocenter – Dynamic Triangle Orthocenter Exploration

POW: Lucky Leon Orthocenter – Dynamic Triangle Orthocenter Exploration
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2017-02-14 06:19
TriangleCoordinate GeometryMatrixTransformationGeometric Construction
① This GeoGebra example dynamically demonstrates the classic theorem that the three altitudes of a triangle intersect at the orthocenter, while also exploring symmetry and matrix-based transformations. ② Core mathematical concepts include the orthocenter, centroid (barycenter), perpendicular lines, symmetry, and a background Poisson distribution tag, involving line equations and matrix operations. ③ Construction details: the Polygon command creates triangle ABC, Distance measures side lengths, OrthogonalLine constructs the altitudes, and PolyLine connects relevant points; objects include vertices A, B, C, triangle poly1, sides a, b, c, altitudes f, g, h, orthocenter D, text labels text1-text5, and lists list1-list3 for dynamic annotations. ④ Suitable for middle/high school geometry or introductory college geometry courses, this interactive tool helps students visualize orthocenter properties, symmetry, and matrix applications in geometry, and can be used for exploratory learning.

GeoGebra Commands & Parameters

  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