解答这道题---**Question 24** (10分) 如图, 在 △ABC 中, ∠CAB = 30°, 点 D 是 AB 右侧一点, DA = DB, DE ⊥ AC 于点 E, 交 AB 于点 F. (1) 如图 1, 若 ∠DBA = 1/2 ∠CAB, AD = 4, 求 DF 的长; (2) 如图 2, 若 ∠C = 90°, 点 G 是 AC 上一点, ∠GBD = 60°, 连接 DG 交 AB 于点 H, 将线段 GH 绕点 G 逆时针旋转 60°, 点 H 恰好落在 DE 上得到点 M, 猜想 AH, DM, BC 之间的数量关系, 并说明理由; (3) 如图 3, 在 (2) 的条件下, ∠ADB = 90°, AG = 4√2, 点 P 是 AB 上一点, 连接 DP, AQ ⊥ DP 于点 Q, 作点 Q 关于 AD 的对称点 R, 请直接写出 BR 的最大值. **Diagram Description:** * **Diagram 1 (图 1):** * Type: Geometric figure. * Main Elements: Triangle ABC. Point D is located outside triangle ABC. Lines AD and BD are drawn. Line segment DE is drawn from D, perpendicular to AC at point E, and intersects AB at point F. Points A, B, C, D, E, F are labeled. * **Diagram 2 (图 2):** * Type: Geometric figure. * Main Elements: A figure including points A, B, C, D, E, F, G, H, M. It appears that ∠C is a right angle. Point G is on AC. Line segment DG intersects AB at H. Line segment DE is perpendicular to AC at E, intersecting AB at F. Point M is on DE. Lines GH and GM are drawn. Points A, B, C, D, E, F, G, H, M are labeled. * **Diagram 3 (图 3):** * Type: Geometric figure. * Main Elements: A figure including points A, B, C, D, E, G, P, Q. It appears that ∠C is a right angle. Point G is on AC. Line segment DE is perpendicular to AC at E. Point P is on AB. Line segment DP is drawn. Line segment AQ is drawn perpendicular to DP at Q. Point R is implied to be the reflection of Q across AD (although R is not labeled in the diagram). Points A, B, C, D, E, G, P, Q are labeled. * **Other Text:** "第 24 题图" is below Diagram 2 and Diagram 3.