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Apollonius' Median Theorem

Apollonius' Median Theorem
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2014-11-10 05:41
TriangleVectorGeometric Construction
This example illustrates Apollonius' median theorem, which states that the length of a median of a triangle is one-third the length of the side to which it is drawn. Key concepts include triangles, medians, and vectors. Using GeoGebra commands such as Segment, Midpoint, Circle, etc., the relationship between the median and the side of the triangle is visually demonstrated. Suitable for students at the middle school level and above for geometric construction.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, A) → c
  4. Midpoint(b) → D
  5. Segment(A, D) → d
  6. Segment(B, D) → e
  7. Segment(D, C) → f
  8. Circle(B, e / 10) → g
  9. OrthogonalLine(B, e) → h
  10. Intersect(g, h) → E_1, F
  11. Line(E_1, e) → i
  12. OrthogonalLine(D, e) → j
  13. Intersect(i, j) → G
  14. Segment(E_1, G) → k
  15. Line(B, C) → l
  16. OrthogonalLine(A, l) → m
  17. Intersect(m, l) → E
  18. Segment(A, E) → n
  19. Segment(B, E) → p
  20. Segment(E, C) → q
  21. Angle(C, E, A) → α
  22. Angle(A, E, B) → β
  23. Angle(C, D, A) → γ
  24. Angle(A, D, B) → δ

AI Drawing Prompt Reference

Draw a triangle ABC with sides AB, BC, and CA. Construct the median from vertex A to the midpoint D of side BC. Then draw segments from B to D and from C to D. Next, draw a circle centered at B with radius equal to BD, and a line through B perpendicular to BD. Find the intersection points E1 and F of the circle and the line. Then draw line segments from E1 to E and from E1 to E. Continue by drawing line BC and a line through A perpendicular to BC. Find the intersection point E of this line with BC. Then draw segments from A to E, from B to E, and from E to C. Finally, measure angles α, β, γ, and δ to illustrate the relationship between the median and the side.
The prompt above was compiled from the original drawing commands and is for reference only.