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Orthocenter and Mirror Points are Concyclic

Orthocenter and Mirror Points are Concyclic
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2015-10-22 00:42
TriangleCircleGeometric Construction
This example demonstrates the concyclicity of the circumcircle formed by the orthocenter, vertex mirror points, and feet of altitudes in a triangle. Key mathematical concepts include orthocenter, reflection, circle, concyclic points, and triangle geometry. GeoGebra commands used in construction are Polygon, OrthogonalLine, Intersect, Mirror, Midpoint, Circle, and Prove. Suitable for high school geometry teaching and exploration to understand advanced triangle properties.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. OrthogonalLine(A, a) → d
  3. OrthogonalLine(C, c) → e
  4. Intersect(d, e) → D
  5. Mirror(D, a) → D'
  6. Midpoint(a) → E
  7. Mirror(D, E) → D'_1
  8. Circle(A, B, C) → f
  9. Prove(AreConcyclic[A, B, C, D'_1]) → g

AI Drawing Prompt Reference

Draw triangle ABC, then construct the perpendiculars from A to sides AB and AC, find their intersection D, reflect D over the midpoint E of AB to get D'1, and draw the circumcircle of triangle ABC. Prove that D'1 lies on this circle.
The prompt above was compiled from the original drawing commands and is for reference only.