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The Gergonne Point and Intouch Triangle

The Gergonne Point and Intouch Triangle
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2015-02-14 18:54
TriangleCirclePolygonGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the Gergonne point of a triangle and its intouch triangle, focusing on the concept of perspective center. Key math concepts include perspective center, intouch triangle, angle bisectors, circles, and polygons. Construction highlights the use of GeoGebra commands like Line, AngularBisector, Intersect, OrthogonalLine, Circle, Segment, and Polygon. Suitable for high school and above students to explore geometric constructions and special points.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. AngularBisector(B, A, C) → d
  5. AngularBisector(C, B, A) → e
  6. AngularBisector(A, C, B) → f
  7. Intersect(d, e) → I
  8. OrthogonalLine(I, c) → g
  9. Intersect(c, g) → D
  10. Circle(I, D) → h
  11. Intersect(h, a) → E, C'
  12. Intersect(h, c) → F, B'
  13. Intersect(h, b) → H, A'
  14. Segment(E, C) → i
  15. Segment(A, H) → j
  16. Segment(B, F) → k
  17. Intersect(i, j) → G_e
  18. Polygon(A', C', B') → poly1, g_1, j_1, k_1

AI Drawing Prompt Reference

Draw a triangle and its three angle bisectors, their intersection point is the Gergonne point. With the Gergonne point as the center, draw a circle with a radius to the intersection of one side, and connect the intersection points with the original vertices of the triangle. Finally, connect these three new points to form the intouch triangle, showing the relationship between the Gergonne point and the intouch triangle.
The prompt above was compiled from the original drawing commands and is for reference only.