帮我生成讲这两道题的视频，并分析这些题目考了哪些章节的知识点---**18. (满分 17 分)** 在长方体 ABCD-A₁B₁C₁D₁ 中, AB = 2, AD = 1, AA₁ = 1, E, F, G, H 分别为 BC, CD, DD₁, D₁A 的中点. (I) 求证: G ∈ 平面 EFH; (II) 点 M 在矩形 A₁B₁C₁D₁ 内 (不含边界) 运动, 若 M ∈ 平面 EFH, 求 M 运动轨迹的长度; (III) 点 O 在矩形 ABCD 内 (不含边界) 运动, 若直线 A₁O // 平面 EFH, 求 A₁O + OC 的最小值. **Chart Description:** * **Type:** Geometric figure (3D cuboid). * **Main Elements:** * **Vertices:** Labeled A, B, C, D, A₁, B₁, C₁, D₁. Vertices of the bottom face are A, B, C, D. Vertices of the top face are A₁, B₁, C₁, D₁. * **Edges:** Edges connect corresponding vertices (e.g., AB, BC, CD, DA, A₁B₁, B₁C₁, C₁D₁, D₁A₁, AA₁, BB₁, CC₁, DD₁). Some edges are represented by solid lines, others by dashed lines, indicating visibility. * **Points:** * E is on edge BC. * F is on edge CD. * G is on edge DD₁. * H is on edge D₁A. * M is marked inside the rectangle A₁B₁C₁D₁. * O is marked inside the rectangle ABCD. * O₁ is marked on the line segment AC. * **Lines/Segments:** * Line segments connect points E, F, G, H, forming a quadrilateral EFGH. * Line segments are drawn from A₁ to M and from O to C. * A line segment A₁O is shown (dashed). * A dashed line segment is shown from A₁ to O₁. * Dashed lines form the cuboid structure. * **Relative Position:** A₁ is above A, B₁ is above B, etc. E is on BC, F on CD, G on DD₁, H on D₁A. M is in the top face, O is in the bottom face. **19. (满分 17 分)** 设 n 次多项式 P_n(x) = a_n x^n + a_{n-1} x^{n-1} + … + a₁ x + a₀ (a_n ≠ 0), 若其满足 P_n(cosθ) = cos(nθ), 则称这些多项式 P_n(x) 为切比雪夫多项式. 例如: 由 cos(1⋅θ) = cosθ 可得切比雪夫多项式 P₁(x) = x; 由 cos(2θ) = 2cos²θ - 1 可得切比雪夫多项式 P₂(x) = 2x² - 1. (I) 求切比雪夫多项式 P₃(x); (II) 请利用 P₃(x) 求 sin 18° 的值. (III) 若 f(x) = 4x³ - 3x - 1/2 在 (-1,1) 内有三个不同的零点 x₁, x₂, x₃, 求 x₁³ + x₂³ + x₃³ 的值. **Other Text:** 第 4 页 共 4 页