题：如图１，设△ＡＢＣ是直角三角形，点Ｄ在斜边ＢＣ上，ＢＤ＝４ＤＣ．已知圆过点Ｃ，且与ＡＣ相交于Ｆ，与ＡＢ相切于ＡＢ的中点Ｇ．求证：ＡＤ⊥ＢＦ．本题为１９９９年全国初中数学联合竞赛第二试第二题，具有一定难度和探索性．本文对此题作如下思考．一、题目的多种新 Question: As shown in Figure 1, set △ABC is a right triangle, point D on the hypotenuse BC, BD = 4DC. It is known that the circle passes through point C and intersects with AC at F and tangent with AB at the midpoint of AB. Proof: AD⊥BF. This question is the second test of the second national junior high school mathematics competition in 1999. It has certain difficulty and exploratory nature. This article considers the following questions. First, a variety of new topics