已知正方形ABCD的边长为6，菱形EFGH的三个顶点E，G，H分别在正方形ABCD边AB，CD，DA上，AH = 2．（1）如图1，当DG = 2，且点F在边BC上时，求证：①△AHE≌△DGH；②菱形EFGH是正方形；（2）如图2，当点F在正方形ABCD的外部时，连接CF．①探究：点F到直线CD的距离是否发生变化？并说明理由；②设DG = x，△FCG的面积为S，是否存在x的值，使得S = 1？若存在，求出x的值；若不存在，请说明理由．---**Image Content Extraction:** **Figure 1 Description:** * **Type:** Geometric figure. * **Main Elements:** * A square ABCD with vertices labeled A (bottom left), B (bottom right), C (top right), and D (top left). * A quadrilateral EFGH is inscribed within the square ABCD. * Point E is located on side AB. * Point F is located on side BC. * Point G is located on side CD. * Point H is located on side DA. * Line segments connect E to F, F to G, G to H, and H to E, forming the quadrilateral EFGH. * Label below the figure: 图1 (Figure 1). **Figure 2 Description:** * **Type:** Geometric figure. * **Main Elements:** * A square ABCD with vertices labeled A (bottom left), B (bottom right), C (top right), and D (top left). * Points E, G, and H are located on sides AB, CD, and DA respectively. * Point E is on side AB. * Point G is on side CD. * Point H is on side DA. * Point F is located outside the square, to the right of side BC. * Line segments connect E to H, H to G, G to F, and F to E, forming the quadrilateral EHGF. * Line segments also connect C to F and B to F, forming triangle CBF adjacent to side BC of the square. * Label below the figure: 图2 (Figure 2). **Other Content:** * No question stem, options, or additional text are present in the image.