题目

 I travel a lot in my work, and what I dislike about my job is eating alone. It always makes me feel lonely to see others laughing and talking. So, room service for several nights was a better choice for me.      After having room service three nights at a hotel in Houston, however, I needed to get out of my room. Although the restaurant opened at 6:30, I arrived at 6:25. The waiter at the front desk made a comment (评论) about my “being there really early”. I explained my dislike of eating alone in restaurants. He then seated me at a lovely table and asked me whether I woul如图，在△ABC中，∠BAC=90°，AD是BC边上的高，E是BC边上的一个动点（不与B，C重合），EF⊥AB，EG⊥AC，垂足分别为F，G．（1）求证：EGAD=CGCD；（2）FD与DG是否垂直？若垂直，请给出证明；若不垂直，请说明理由；（3）当AB=AC时，△FDG为等腰直角三角形吗？并说明理由．