Implicit loci

Zoltán Kovács, May 13, 2016

GeoGebra's [b]LocusEquation[/b] command allows the automatic discovery of loci of points in diagrams defined by implicit conditions. This represents an extension of the former (explicit) command, similar in nature, but restricted to loci defined by the standard mover-tracer construction. The new syntax automates the ‘dummy locus dragging’ in dynamic geometry. This collection was compiled by a group of GeoGebra contributors, namely [i]Miguel Á. Abánades[/i], [i]Francisco Botana[/i], [i]Zoltán Kovács[/i], [i]Tomás Recio[/i] and [i]Csilla Sólyom-Gecse[/i].


  • 1. Discovering the Simson-Wallace theorem
  • 2. A generalization of the right triangle altitude theorem
  • 3. Dependency of locus sets on the initial construction 1
  • 4. Dependency of locus sets on the initial construction 2
  • 5. Degenerate cases and necessary not-sufficient conditions
  • 6. Holfeld's 35th problem as an implicit locus
  • 7. A necessary not-sufficient condition

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Discovering the Simson-Wallace theorem

[list=1][*]Given the triangle ABC.[br][/*][*]Construct the lines of the sides.[/*][*]Let D be an arbitrary point.[/*][/list]In Steps 4-9 we create the perpendicular projection of D to the lines constructed in Step 2.[br]Is it possible to choose D in order to be the projection points (feet points E, F and G) aligned?[br]In Step 10 GeoGebra solves this question by computing [code]LocusEquation[AreCollinear[E,F,G],D][/code].
– Zoltán Kovács

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