这是一道初中数学题，请分析其中的知识点，同时深入浅出，逐步引导学生得到详细解题步骤，思路要简洁清晰，使用中文讲解---**Question 21** **Problem Description:** 如图, ⊙O 是 △ABC 的外接圆, BE 为 ⊙O 的直径, BE 与 AC 交于点 F, D 为 BE 延长线上一点, 连接 CD, CE, AE, ∠BAC + ∠BCD = 180°. As shown in the figure, ⊙O is the circumcircle of △ABC, BE is the diameter of ⊙O, BE intersects AC at point F, D is a point on the extension of BE, connect CD, CE, AE, ∠BAC + ∠BCD = 180°. **(1) 求证: ∠DCE = ∠CBD;** (1) Prove: ∠DCE = ∠CBD; **(2) 若 AB = BC, tan D = 4/3, ⊙O 半径为 4, 求 BC 的长.** (2) If AB = BC, tan D = 4/3, the radius of ⊙O is 4, find the length of BC. **Chart Description:** * **Type:** Geometric diagram involving a circle and a triangle. * **Main Elements:** * **Circle:** A circle with center O. * **Points:** Labeled points A, B, C are on the circle. Point O is at the center. Point F is the intersection of BE and AC. Point E is on the circle. Point D is on the line extending BE. * **Lines/Segments:** Line segments AB, BC, AC form triangle ABC inscribed in the circle. BE is a line segment passing through O and connecting points B and E on the circle (diameter). Line segment AC intersects BE at F. Line segment BE is extended to point D. Line segments CD, CE, AE are drawn. * **Relationships:** BE is a diameter of the circle. O is the center. F lies on both BE and AC. D lies on the extension of BE beyond E. ∠BAC and ∠BCD are angles shown in the figure. * **Labels:** Points O, A, B, C, D, E, F are labeled.