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直角三角形中,斜边上的中线、高、直角平分线都是重要线段,它们都与三角形的三边有着密切的关系。其中中线和高与三边的关系十分明显,无需赘述,而直角平分线与三边的关系并非人人皆知。初中几何教材中未用定理的形式表述过直角平分线与三边的关系,也没有用适当的习题给予启发和引导。相反,教材中却有一道例题(《几何》第二册P.46),用了很繁琐的方法去求直角平分线的长。这于教于学都很不利。
In a right-angled triangle, the midline, high, and right angle bisectors on the hypotenuse are all important line segments. They all have a close relationship with the three sides of the triangle. The relationship between the midline and the high and the three sides is very obvious and there is no need to go into details. The relationship between the right angle bisector and the three sides is not known. The form of the unused theorem in junior school geometry textbooks describes the relationship between the right angle bisector and the three sides, and does not give inspiration and guidance with appropriate exercises. On the contrary, there is an example in the textbook (Geometry, second volume, P.46), which uses a very tedious method to determine the length of the right angle bisector. This is not good for teaching.