Ellipse as a Cone Section, Geometric Proof

GeoGebra Commands & Parameters

1. Intersect(yAxis, xAxis) → A
2. Point(zAxis) → B
3. Cone(A, B, 1) → a_{5}, k_1, a_{4}
4. Point(yAxis) → C
5. Point(zAxis) → D
6. Line(D, xAxis) → f
7. Point(f) → E
8. Circle(zAxis, D) → c_{2}
9. Circle(zAxis, E) → d
10. Intersect(f, d, 2) → F
11. Plane(F, E, C) → p
12. PointIn(p) → E_1
13. PointIn(p) → G_1
14. PointIn(p) → H_1
15. PointIn(p) → I_1
16. Polygon(E_1, G_1, H_1, I_1) → v2, b, f_1, m_1, b_1
17. IntersectPath(p, a_{5}) → e
18. Intersect(f, e, 1) → G
19. Intersect(f, e, 2) → P_3
20. Intersect(yAxis, k_1, 2) → I
21. Segment(B, I) → g
22. Intersect(g, e) → S_2,
23. Intersect(yAxis, k_1, 1) → K
24. Segment(B, K) → h
25. Intersect(h, e) → L,
26. Segment(S_2, L) → i
27. Midpoint(S_2, L) → M
28. Line(M, f) → j
29. Point(j) → N
30. Point(j) → O
31. Segment(N, O) → k_{4}
32. Segment(S_2, I) → l
33. Segment(L, K) → m
34. Intersect(k_{4}, e) → S_4, S_3
35. Line(S_2, zAxis) → n
36. Line(L, zAxis) → q
37. Line(M, q) → r
38. Intersect(yAxis, r) → R_{1}
39. Intersect(yAxis, n) → S
40. Intersect(yAxis, q) → T
41. Circle(r, S) → s
42. Line(L, yAxis) → t
43. Segment(R_{1}, S) → c
44. Intersect(r, t) → U
45. Cylinder(U, R_{1}, c) → o, c_1, d_1, u
46. Segment(M, S_4) → b_{MR}
47. Point(r) → V
48. Cylinder(V, R_{1}, b_{MR}) → v, e_1, g_1, w
49. IntersectPath(p, v) → h_1
50. Segment(S_2, I) → i_1
51. Segment(L, K) → j_1
52. IntersectPath(p, o) → k_{3}
53. Corner(k_{3}) → S_1, Z
54. Intersect(yAxis, d_1, 2) → A_1
55. Segment(A_1, R_{1}) → l_1
56. Intersect(k_{4}, r) → M_e
57. Circle(r, M_e) → p_1
58. Circle(r, M_e) → q_1
59. Circle(r, M_e) → r_1
60. Circle(M_e, l_1, r) → s_1
61. Segment(M_e, S_1) → a
62. OrthogonalLine(S_3, p) → n_1
63. Circle(S_3, a, n_1) → k_{2}
64. Intersect(a, k_{2}) → F_1,
65. Mirror(F_1, k_{4}) → F_2
66. Segment(S_3, F_1) → a_{2}
67. Segment(S_3, F_2) → a_1
68. Point(e) → P
69. Segment(F_1, P) → f_{PF_1}
70. Segment(P, F_2) → f_{PF2}
71. Point(e) → P'
72. Segment(F_1, P') → f_{P'F1}
73. Segment(P', F_2) → f_{P'F2}
74. Point(k_{2}) → k
75. Segment(I, K) → t_1
76. Line(A, r) → j_2
77. PointIn(p) → H
78. PointIn(p) → B_1
79. PointIn(p) → C_1
80. PointIn(p) → D_1