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Proof of Two Times the Squares of Two Numbers with a Difference of One Equals On

Proof of Two Times the Squares of Two Numbers with a Difference of One Equals On
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2016-12-09 05:20
FunctionCoordinate GeometryGeometric Construction
This example demonstrates the geometric proof of the algebraic identity that two times the squares of two numbers with a difference of one is equal to one plus the sum of those two numbers squared. It involves core mathematical concepts such as algebraic identity proof, square operations, and algebraic manipulation. The construction uses GeoGebra commands/objects like Line, Point, Segment, OrthogonalLine, Circle, Intersect, and Polygon. It is suitable for high school students to understand algebraic identities and their geometric interpretations, as well as for math competition preparation.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → f
  2. Point(Circle[A, a]) → B
  3. Segment(A, B) → g
  4. OrthogonalLine(A, f) → h
  5. Circle(A, B) → c
  6. Intersect(c, h, 2) → C
  7. OrthogonalLine(B, f) → i
  8. OrthogonalLine(C, h) → j
  9. Intersect(i, j) → D
  10. Point(Circle[B, a]) → E
  11. Segment(B, E) → k
  12. OrthogonalLine(E, f) → l
  13. Intersect(j, l) → F
  14. Point(Circle[C, a + 1]) → G
  15. Segment(C, G) → m
  16. Circle(C, G) → d
  17. Intersect(d, h, 2) → H
  18. OrthogonalLine(G, j) → n
  19. OrthogonalLine(H, n) → p
  20. Point(Circle[G, a + 1]) → I
  21. Segment(G, I) → q
  22. OrthogonalLine(I, j) → r
  23. Intersect(p, r) → J
  24. Intersect(n, p) → K
  25. Segment(A, B) → s
  26. Segment(B, D) → t
  27. Segment(D, C) → b
  28. Segment(C, A) → e
  29. Segment(B, E) → f_1
  30. Segment(E, F) → g_1
  31. Segment(F, D) → h_1
  32. Segment(C, H) → i_1
  33. Segment(H, K) → j_1
  34. Segment(K, G) → k_1
  35. Segment(K, J) → l_1
  36. Segment(J, I) → m_1
  37. Circle(C, D) → c_1
  38. Intersect(c_1, i_1) → L,
  39. OrthogonalLine(L, i_1) → n_1
  40. Intersect(i, n_1) → M
  41. Intersect(l, n_1) → N
  42. Segment(D, M) → p_1
  43. Segment(M, L) → q_1
  44. Segment(M, N) → r_1
  45. Segment(N, F) → s_1
  46. Intersect(n_1, m_1) → O
  47. Intersect(l, l_1) → P
  48. Segment(N, P) → t_1
  49. Segment(N, O) → a_1
  50. Polygon(A, B, D, C) → poly1, a_2, b_1, d_1, c_2
  51. Polygon(B, E, F, D) → poly2, b_2, e_1, f_2, d_2
  52. Polygon(F, N, M, D) → poly3, f_3, n_2, m_2, d_3
  53. Polygon(D, C, L, M) → poly4, d_4, c_3, l_2, m_3
  54. Segment(L, H) → g_2
  55. Polygon(F, I, O, N) → poly5, f_4, i_2, o, n_3
  56. Polygon(N, L, H, P) → poly6, n_4, l_3, h_2, p_2
  57. Polygon(N, P, J, O) → poly7, n_5, p_3, j_2, o_1
  58. Segment(D, G) → k_2

AI Drawing Prompt Reference

Draw a coordinate plane with a horizontal baseline through the origin. Place a point on a circle centered at the origin with a given radius, and connect it to the origin. Construct a perpendicular line through the origin to the baseline, and intersect it with a circle centered at the origin through the first point to get a second point. From this second point, draw a line parallel to the baseline, and from the first point draw a perpendicular to the baseline; their intersection forms a rectangle. Repeat a similar construction on the other side to create a second rectangle, connecting vertices to form a larger polygon. Finally, draw a circle centered at one vertex with radius equal to an adjacent side, intersect it with a rectangle side, and add perpendicular and parallel lines to form smaller rectangles, illustrating the relationship between the sum of two squares and another square.
The prompt above was compiled from the original drawing commands and is for reference only.