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Six Orthocenters on One Circle

Six Orthocenters on One Circle
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2021-02-11 18:45
TriangleCirclePolygonGeometric ConstructionLocus
This example demonstrates a striking geometric fact: under a specific configuration, six orthocenters of related triangles all lie on a single circle (an orthocenter concyclicity phenomenon). The key math ideas are triangle orthocenters and concyclicity via circle geometry. The construction uses GeoGebra commands revealed in the construct summary: Circle to define the circle, Point to create key points, TriangleCenter to compute various triangle centers/orthocenters, Segment together with Tangent and Intersect to build auxiliary lines, tangents, and intersection points that determine the underlying triangles, and finally AreConcyclic to verify that the selected six orthocenters are concyclic. Suitable for geometry construction practice, advanced high-school olympiad preparation, and dynamic geometry exploration.

GeoGebra Commands & Parameters

  1. Circle(I, 3) → c_{2}
  2. Point(Circle[I, 0.6]) → B_{7}
  3. Segment(I, B_{7}) → f_{1}
  4. Circle(B_{7}, 3.724734950228972) → d_{7}
  5. Point(d_{7}) → A_{6}
  6. Tangent(A_{6}, c_{2}) → g, h
  7. Intersect(d_{7}, g, 2) → A_{1}
  8. Intersect(d_{7}, h, 2) → A_{5}
  9. Tangent(A_{5}, c_{2}) → i, j
  10. Intersect(d_{7}, j, 2) → A_{4}
  11. Tangent(A_{4}, c_{2}) → k, l
  12. Intersect(d_{7}, l, 2) → A_{3}
  13. Tangent(A_{3}, c_{2}) → m, n
  14. Intersect(d_{7}, n, 2) → A_{2}
  15. Tangent(A_{2}, c_{2}) → p_{1}, q_{1}
  16. Polygon(A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}) → poly1, a, b, c, d, e, f
  17. Circle(A_{1}, A_{2}, A_{3}) → p
  18. Intersect(c_{2}, a) → B_1, B
  19. Intersect(c_{2}, p) → C, , ,
  20. Intersect(c_{2}, b) → B_2,
  21. Intersect(c_{2}, c) → B_3,
  22. Intersect(c_{2}, d) → B_4,
  23. Intersect(c_{2}, e) → B_5,
  24. Intersect(c_{2}, f) → B_{6},
  25. Polygon(B_{6}, B_1, B_2, B_3, B_4, B_5) → poly2, b_6, b_1, b_2, b_3, b_4, b_5
  26. TriangleCenter(B_{6}, A_{1}, B_1, 4) → H_1
  27. TriangleCenter(B_1, A_{2}, B_2, 4) → H_2
  28. TriangleCenter(B_2, A_{3}, B_3, 4) → H_3
  29. TriangleCenter(B_3, A_{4}, B_4, 4) → H_4
  30. TriangleCenter(B_4, A_{5}, B_5, 4) → H_5
  31. TriangleCenter(B_5, A_{6}, B_{6}, 4) → H_6
  32. Circle(H_1, H_2, H_3) → q
  33. AreConcyclic(H_1, H_2, H_3, H_4) → o
  34. AreConcyclic(H_1, H_2, H_3, H_5) → r
  35. AreConcyclic(H_1, H_2, H_3, H_6) → s

AI Drawing Prompt Reference

Draw a base circle and place a moving point inside it, connecting the center to this point to form a radius. Use this moving point as the center to draw another circle, and pick a point on it. From this point, draw two tangents to the base circle; the intersections of these tangents with the auxiliary circle sequentially determine six points forming a hexagon. Then, using the intersections of the base circle with each side of this hexagon, form a second hexagon. Construct six triangles, each using two adjacent vertices of the first hexagon and the corresponding intersection point on the base circle, and find the orthocenter of each triangle. Finally, verify that these six orthocenters are concyclic and draw the circle passing through the first three orthocenters to illustrate this concyclicity.
The prompt above was compiled from the original drawing commands and is for reference only.