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Similar Triangles Task 3: Dynamic Construction and Incircle Exploration

Similar Triangles Task 3: Dynamic Construction and Incircle Exploration
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2017-11-21 10:43
TriangleCircleTransformationGeometric ConstructionSlider Animation
In collections Similar Triangles Tranformations
This GeoGebra example demonstrates the dynamic construction of a set of similar triangles, incorporating incircles, rotations, and orthogonal lines to explore the properties and proportional relationships of similar triangles. The core mathematical concepts include corresponding sides being proportional, corresponding angles being equal, and the geometric significance of the incircle (inscribed circle). Key construction steps involve using the Circle command to draw circles, Intersect to find intersection points, Polygon to construct polygons, Segment to connect points, Point to define key vertices, Angle to measure angles, Rotate to transform figures, OrthogonalLine to create perpendiculars, Midpoint to locate midpoints, Line to draw straight lines, and Textfield for interactive input. The objects include multiple triangles (poly2, poly3, poly4) and their scaled versions (marked with primes), as well as incircle-related elements (i, p, G). This example is suitable for upper middle school or high school students, serving as a visual aid for understanding the criteria and properties of similar triangles, and as an advanced exercise in geometric construction using GeoGebra's dynamic tools.

GeoGebra Commands & Parameters

  1. Circle(A', Segment[A', B']) → h
  2. Circle(D', Segment[D', B']) → k
  3. Intersect(h, k, 2) → F
  4. Polygon(B', A', F, D') → poly2, b', a', f_1, d'
  5. Segment(A', D') → i
  6. Circle(A', 2.5) → p
  7. Point(p) → G
  8. Segment(G, A') → j
  9. Angle(D', A', G) → Θ
  10. Rotate(B', Θ - 180°, A') → B''
  11. Rotate(A', Θ - 180°, A') → A''
  12. Rotate(F, Θ - 180°, A') → F'
  13. Rotate(D', Θ - 180°, A') → D''
  14. Polygon(B'', A'', F', D'') → poly2', b'', a'', f', d''
  15. Polygon(B'', A', D'') → poly3, d''_1, b''_1, a'_1
  16. Polygon(A', F', D'') → poly4, d''_2, a'_2, f'_1
  17. OrthogonalLine(A', b''_1) → l
  18. Point(l) → H
  19. Segment(H, A') → m
  20. Angle(D'', A', B'') → δ
  21. Angle(B'', D'', A') → ζ
  22. Midpoint(d''_1) → I
  23. Midpoint(a'_1) → J
  24. Midpoint(b''_1) → K
  25. Midpoint(d''_2) → L
  26. Midpoint(a'_2) → M
  27. Angle(A', D'', F') → θ
  28. Line(A', xAxis) → b_1
  29. Intersect(p, b_1, 1) → O
  30. Angle(F', A', D'') → ι
  31. Textfield(scale) → inputBox1
  32. Polygon(A_0, B_0, C_0) → poly5, c_1, a_0, b_0
  33. Angle(A_0, B_0, C_0) → μ
  34. Polygon(X, Y, Z) → poly6, p_1, c_2, n