Derivation and Visualization of Ellipse Area Formula

GeoGebra Commands & Parameters

1. Segment(O, Z) → a
2. Point(a) → A
3. Mirror(A, O) → A'
4. Circle(O, Distance[O, A]) → c
5. OrthogonalLine(O, a) → b
6. Intersect(c, b, 2) → D
7. Intersect(c, b, 1) → E
8. Point(b) → B
9. Ray(O, A') → d
10. Circle(B, Segment[O, A]) → e
11. Intersect(e, a, 1) → F_1
12. Intersect(e, d, 1) → F_2
13. Ellipse(F_2, F_1, B) → f
14. Sequence(((i * x(A) / n), 0), i, 0, n) → x_{i}
15. Sequence(OrthogonalLine[Element[x_{i}, i], a], i, 1, n) → lines_{vert}
16. Sequence(Intersect[f, Element[lines_{vert}, i], 2], i, 1, n) → y_{ellip}
17. Sequence(Intersect[c, Element[lines_{vert}, i], 2], i, 1, n) → y_{circ}
18. Sequence(Polygon[{Element[x_{i}, 1 + j], Element[y_{ellip}, 1 + j], Element[y_{ellip}, 2 + j], Element[x_{i}, 2 + j]}], j, 0, n - 2) → traps_{low}
19. Polygon(Element[x_{i}, n], Element[y_{ellip}, n], A) → poly1, g, h, i
20. Polygon(Element[x_{i}, n], Element[y_{circ}, n], A) → poly2, a_1, j, k
21. Sequence(Polygon[{Element[x_{i}, 1 + j], Element[y_{circ}, 1 + j], Element[y_{circ}, 2 + j], Element[x_{i}, 2 + j]}], j, 0, n - 2) → traps_{high}
22. OrthogonalLine(X_1, a) → p
23. OrthogonalLine(X_2, a) → q
24. Intersect(f, p, 2) → Y_{1e}
25. Intersect(c, p, 2) → Y_{1c}
26. Intersect(f, q, 2) → Y_{2e}
27. Intersect(c, q, 2) → Y_{2c}
28. Polygon(X_1, Y_{1e}, Y_{2e}, X_2) → poly3, x_1, c_1, g_1, x_2
29. Polygon(X_1, Y_{1c}, Y_{2c}, X_2) → poly4, x_3, f_1, h_1, x_4
30. Segment(Y_{1e}, Y_{1c}) → r
31. Segment(Y_{2e}, Y_{2c}) → s
32. Point(r) → I
33. Point(s, PathParameter[I]) → J
34. Polygon(X_1, I, J, X_2) → poly5, x_5, i_1, j_1, x_6
35. PathParameter(I) → o
36. Sequence(Segment[Element[y_{ellip}, i], Element[y_{circ}, i]], i, 1, n) → segs
37. Sequence(Point[Element[segs, i], o], i, 1, n) → y_{slide}
38. Sequence(Segment[Element[x_{i}, j], Element[y_{slide}, j]], j, 1, n) → big_{segs}
39. Append(Append[y_{slide}, A], O) → list1
40. Polygon(list1) → poly6
41. Vector(I, (x(I), y(I) - (0.25 * (x(A) - y(B))))) → v
42. Vector(I, (x(I), y(I) + (0.25 * (x(A) - y(B))))) → w
43. Segment(O, A) → b_1
44. Segment(O, B) → d_1
45. Conic(Element[y_{slide}, 1], A, A', (-Element[y_{slide}, 1]), Element[y_{slide}, 2]) → e_1