对题目进行解答---**1. (本题 12 分)** 如图 1, 四边形 ABCD 内接于 ⊙O, BD 为直径, ∠ABC 为锐角, 过点 B 作 BE ⊥ AC 于点 E, 过点 A 作 BC 的平行线交 BE 的延长线于点 F. (1) ∠ABD = a, 请用含 a 的代数式表示 ∠CBE. (2) 若 AF = BD, 求证: AD = AE. (3) 如 图 2, 在 (2) 的条件下, BF 与 ⊙O 交于点 G, 与 AD 延长线交于点 H, 连接 DG. ① 若 CD = 4, DG = 1, 求 AD 的长. ② 若 cos∠AHB = DG/HF, 求 tan∠ABD 的值. **Diagram Description:** **图 1:** * Type: Geometric diagram. * Main Elements: * A circle with center O. * Four points A, B, C, D are on the circumference of the circle, forming an inscribed quadrilateral ABCD. * The line segment BD is a diameter passing through the center O. * Point E is located on the line segment AC. * A line segment BE is drawn, perpendicular to AC at E (indicated by a right angle symbol at E). * The line segment BE is extended beyond E to a point F. * A line segment AF is drawn. * The line segment AF is parallel to the line segment BC (indicated by parallel lines annotations in the problem text). * Lines connecting the points: AB, BC, CD, DA, AC, BD, BE, EF, AF. * The problem states ∠ABC is an acute angle. * Label: 图 1 **Diagram for Question (3) (referred to as 图 2 in the text):** * Type: Geometric diagram. * Main Elements: * This diagram is based on 图 1 with additional elements and conditions from part (2) and (3). * It includes the circle with center O and inscribed quadrilateral ABCD, BD as diameter, BE ⊥ AC at E, and the line B-E-F where AF || BC. * The line segment BF intersects the circle at point G (apart from B). * The line segment AD is extended to a point H. * The line segment BF intersects the extension of AD at point H. * A line segment DG is drawn. * Lines connecting the points: AB, BC, CD, DA (extended to H), AC, BD, BE, EF, AF, BG, GF, AH, DH, GH, DG. * Points G and H are explicitly labeled.