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平面几何中证明两直线互相垂直的方法过多,但用线段运算来证明两直线互相垂直的方法,却没有介绍。笔者有感于此,作成下文,供初中数学教师教学时参考。定理若平面上四点P、Q、M、N满足关系式 PM~2-PN~2=QM~2-QN~2。*则直线PQ与MN互相垂直。证明不难证明,P、Q、M、N若满足关系式(*),则直线PQ与MN不会平行,也不会重合,它们一定相交。按照交点同在二线段上;在其一上而在另一线段延长线上;都在二线段延长线上,可分三种情况。也就是说,同平面上的四点P、Q、M、N连
There are too many ways to prove that two straight lines are perpendicular to each other in plane geometry, but the method of using line segment operations to prove that two straight lines are perpendicular to each other is not introduced. The author felt this, made the following, for reference when junior middle school math teacher teaching. Theorem If the four points on the plane P, Q, M, N satisfy the relationship PM~2-PN~2=QM~2-QN~2. * The straight line PQ and MN are perpendicular to each other. It is not difficult to prove that if P, Q, M, and N satisfy the relation (*), the straight line PQ and MN will not be parallel or coincident, and they must intersect. According to the intersection point, they are on the second line segment; on one line they are on the extension line of the other line segment; on the extension line of the second line segment, they can be divided into three situations. That is, four points P, Q, M, and N in the same plane