Eccentricity Proof

GeoGebra Commands & Parameters

1. If(abs(Z_{int}) > E_{psilon} ∧ abs(sin(θ_{plane-r})) < 0.9999, true, false) → S_{tandard}
2. Line(P_{z}, (1, 0, z(P_{z}) + tan(θ_{plane-r}))) → L_{plane}
3. Line((0, 0, 0), (r_{cone}, 0, 5)) → L_{cone-r}
4. AngularBisector(L_{plane}, L_{cone-r}) → L_{bis-1}, L_{bis-2}
5. Intersect(L_{bis-1}, zAxis) → P_{circle-1}
6. OrthogonalLine(P_{circle-1}, L_{plane}, x = 0) → L_{perp-1}
7. If(S_{tandard}, Intersect[L_{perp-1}, L_{plane}]) → F_{1}
8. OrthogonalLine(P_{circle-1}, L_{cone-r}, x = 0) → L_{perp-1a}
9. Intersect(L_{perp-1a}, L_{cone-r}) → P_{horz-1}
10. Line(P_{horz-1}, xAxis) → L_{horz-1}
11. Intersect(L_{horz-1}, L_{plane}) → P_{d1}
12. Intersect(L_{bis-2}, zAxis) → P_{circle-2}
13. OrthogonalLine(P_{circle-2}, L_{plane}, x = 0) → L_{perp-2}
14. If(S_{tandard}, Intersect[L_{perp-2}, L_{plane}]) → F_{2}
15. If(S_{tandard}, Distance[P_{circle-2}, L_{plane}], sqrt(-1)) → r_{2}
16. Circle(P_{circle-2}, r_{2}, y = 0) → C_{circle-2}
17. If(S_{tandard}, Distance[P_{circle-1}, L_{plane}], sqrt(-1)) → r_{1}
18. Circle(P_{circle-1}, r_{1}, y = 0) → C_{circle-1}
19. Line((0, 0, 0), ((-r_{cone}), 0, 5)) → L_{cone-l}
20. Cone((0, 0, 5), (0, 0, 0), r_{cone}) → C_{cone-top}, C_1, C_3
21. Cone((0, 0, -5), (0, 0, 0), r_{cone}) → C_{cone-bot}, c, C_2
22. Intersect(L_{plane}, L_{cone-r}) → P_{r}
23. Intersect(L_{plane}, L_{cone-l}) → P_{l}
24. Sphere(P_{circle-1}, r_{1}) → s_{1}
25. Sphere(P_{circle-2}, r_{2}) → s_{2}
26. OrthogonalLine(P_{circle-2}, L_{cone-r}, x = 0) → L_{perp-2a}
27. Intersect(L_{perp-2a}, L_{cone-r}) → P_{horz-2}
28. Line(P_{horz-2}, xAxis) → L_{horz-2}
29. Intersect(L_{horz-2}, L_{plane}) → P_{d2}
30. CurveCartesian((((r(θ) * sin(θ)) * cos(θ_{plane-r})) * sgn(Z_{int+e})) + x(F_{1}), (r(θ) * cos(θ)), (((r(θ) * sin(θ)) * sin(θ_{plane-r})) * sgn(Z_{int+e})) + z(F_{1}), θ, 0, 6.3) → C_{conic}
31. If(S_{tandard}, C_{conic}) → C_{conic-disp}
32. If(¬S_{tandard}, Line[(0, 0, 0), (cot(θ_{plane-r}), (-y_{deg}), 1)]) → L_{degm}
33. If(¬S_{tandard}, Line[(0, 0, 0), (cot(θ_{plane-r}), y_{deg}, 1)]) → L_{degp}
34. If(S_{tandard}, Line[P_{d1}, yAxis]) → L_{d1}
35. If(S_{tandard}, Line[P_{d2}, yAxis]) → L_{d2}
36. If(S_{tandard}, C_{conic}) → a
37. OrthogonalLine(P, L_{d1}, space) → L_{PD-line}
38. Intersect(L_{PD-line}, L_{d1}) → D
39. Segment(P, D) → L_{PD}
40. Segment(P, F_{1}) → L_{PF}
41. IntersectPath(P_{plane1}, s_{1}) → C_{s1-plane1}
42. IntersectPath(P_{plane2}, s_{2}) → C_{plane2-s2}
43. OrthogonalLine(P, P_{plane1}) → L_{P-plane1}
44. Intersect(L_{P-plane1}, P_{plane1}) → A
45. Segment(P, A) → L_{PA}
46. Segment(D, A) → L_{DA}
47. Plane(F_{1}, P, A) → P_{PFA}
48. Line(P, (0, 0, 0)) → L_{P-0}
49. Intersect(L_{P-0}, P_{plane1}) → F'
50. Segment(P, F') → L_{PF'}
51. Segment(F', A) → L_{F'A}
52. Ray(D_g, (100, y(D_g))) → f
53. Circle(D_g, L_{DA}) → d
54. Intersect(d, f) → A_{g-plane},
55. Circle(A_{g-plane}, L_{PA}) → h
56. Ray(A_{g-plane}, (x(A_{g-plane}), 100)) → g
57. Intersect(h, g) → P_{g-plane},
58. Segment(D_g, A_{g-plane}) → i
59. Segment(A_{g-plane}, P_{g-plane}) → s
60. Segment(P_{g-plane}, D_g) → t
61. Angle(A_{g-plane}, D_g, P_{g-plane}) → θ_{plane}
62. Ray(F'_g, (100, y(F'_g))) → j
63. Circle(F'_g, L_{F'A}) → k
64. Intersect(k, j) → A_{g-cone},
65. Ray(A_{g-cone}, (x(A_{g-cone}), 100)) → l
66. Circle(F'_g, L_{PF'}) → p
67. Intersect(p, l) → P_{g-cone},
68. Segment(F'_g, A_{g-cone}) → m
69. Segment(A_{g-cone}, P_{g-cone}) → n
70. Segment(P_{g-cone}, F'_g) → q
71. Angle(A_{g-cone}, F'_g, P_{g-cone}) → θ_{cone}