讲解这道题---**1. Question Stem:** 14. 如图 (As shown in the figure), 正方形 ABCD 的边长为 a (the side length of square ABCD is a), 点 E 在边 AB 上运动 (不与点 A, B 重合) (point E moves on side AB, not coinciding with points A or B), ∠DAM = 45°, 点 F 在射线 AM 上 (point F is on ray AM), 且 AF = √2BE (and AF = √2BE), CF 与 AD 相交于点 G (CF intersects AD at point G), 连接 EC、EF、EG (connect EC, EF, EG). 则 (Then): **2. Conclusions (Options to evaluate):** 下列结论 (The following conclusions): ① ∠ECF = 45° ② △AEG 的周长为 (1 + (√2)/2)a (The perimeter of △AEG is (1 + (√2)/2)a) ③ BE² + DG² = EG² ④ 当 BE = (1/3)a 时, G 是线段 AD 的中点 (When BE = (1/3)a, G is the midpoint of segment AD) **3. Other Relevant Text:** 其中正确的结论是 ①④ (Among them, the correct conclusions are ①④). *(Note: The conclusions ② and ③ have strike-through marks on the original image, and ① and ④ are circled and indicated as correct in the final part of the question. This implies that the provided image includes the question along with a pre-marked answer.)* **4. Chart/Diagram Description:** * **Type:** The problem statement begins with "如图" (as shown in the figure), indicating that a geometric figure is expected to accompany this problem. * **Main Elements:** No figure is provided in the image for description. Based on the textual description, the implied figure would involve: * A square ABCD with side length 'a'. * Point E on side AB. * A ray AM originating from A, forming ∠DAM = 45°. * Point F on ray AM. * Line segment CF intersecting AD at point G. * Line segments EC, EF, EG connecting the points. * Specific length and angle relations: ∠DAM = 45°, AF = √2BE.