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Exploring the Area of a Triangle by Anthony OR 柯志明

Exploring the Area of a Triangle by Anthony OR 柯志明
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2024-05-23 10:21
TriangleAngleVectorTrigonometryGeometric Construction

GeoGebra Commands & Parameters

  1. Point(Circle[O, sqrt(101)]) → A
  2. Angle(O - (10, 1), O, A) → θ
  3. Mirror(A, Rotate[x = x(O), θ, O]) → B
  4. Intersect(Circle[A, 7], Circle[B, 15], 1) → C
  5. Polygon(O, A, B, C, A) → t1, f, g, h, i, j
  6. Circle(O, sqrt(101)) → d
  7. Polygon(A, B, C) → t, c, a, b
  8. Point(Circle[O_{1}, sqrt(101)]) → A_{1}
  9. Angle(O_{1} - (10, 1), O_{1}, A_{1}) → θ_{1}
  10. Mirror(A_{1}, Rotate[x = x(O_{1}), θ_{1}, O_{1}]) → B_{1}
  11. Intersect(Circle[A_{1}, 7], Circle[B_{1}, 15], 1) → C_{1}
  12. Polygon(O_{1}, A_{1}, B_{1}, C_{1}, A_{1}) → t1_{1}, f_{1}, g_{1}, h_{1}, i_{1}, j_{1}
  13. Polygon(A_{1}, B_{1}, C_{1}) → t2, a_1, b_1, c_1
  14. Point(Circle[D, 1]) → E
  15. Polygon(D, E, F, G) → q1, d_1, e, f_2, g_2
  16. UnitVector(Segment[D, E]) → u
  17. UnitOrthogonalVector(Segment[D, E]) → v
  18. Sequence(Segment[D + (i * u), D + (i * u) + (0.6 * v)], i, 1, 20) → l1
  19. Sequence(Segment[D + (i * u) + (0.5 * u), D + (i * u) + (0.5 * u) + (0.5 * v)], i, 0, 19) → l2
  20. Sequence(Segment[D + (i * v), D + (i * v) + (0.6 * u)], i, 1, 20) → l3
  21. Sequence(Segment[D + (i * v) + (0.5 * v), D + (i * v) + (0.5 * u) + (0.5 * v)], i, 0, 19) → l4
  22. Sequence(Segment[D + ((0.1 * i) * u), D + ((0.1 * i) * u) + ((0.1 * i) * v)], i, 1, 4) → l5
  23. Sequence(Segment[D + ((0.1 * i) * v), D + ((0.1 * i) * v) + ((0.1 * i) * u)], i, 1, 4) → l6
  24. Sequence(Sequence[Segment[D + ((0.5 * i) * u) + ((0.1 * j) * u) + (0.5 * u), D + ((0.5 * i) * u) + ((0.1 * j) * u) + (0.5 * u) + (0.4 * v)], i, 0, 38], j, 1, 4) → m1, {{0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999}, {0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999}, {0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999}, {0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999, 0.3999999999999999}}
  25. Sequence(Sequence[Segment[D + ((0.5 * i) * v) + ((0.1 * j) * v) + (0.5 * v), D + ((0.5 * i) * v) + ((0.1 * j) * v) + (0.4 * u) + (0.5 * v)], i, 0, 38], j, 1, 4) → m2, {{0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986}, {0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986}, {0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986}, {0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986, 0.3999999999999986}}

AI Drawing Prompt Reference

Draw a triangle with point O as the center and points A and B as vertices. First, take point A on the circle centered at O. Then, find point B using the angle θ between points A and O. Construct triangle t1 with points O, A, and B. Next, find point C by intersecting a circle centered at A with radius 7 and a circle centered at B with radius 15. Construct triangle t with points O, A, B, and C. Repeat the process to construct triangle t2 with points O1, A1, and B1. Finally, take point E on a circle centered at D with radius 1, and construct a grid of parallel and perpendicular lines through points D and E.
The prompt above was compiled from the original drawing commands and is for reference only.