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上海市中学教师进修教材《初等几何复习与研究》上册第127页编有这么一个命题:“已知正方形 ABCD 内部有一个小正方形 EFGH,而 M、N、P、Q 分别为 AH、BE、CF、DG 的中点,则 MNPQ 是正方形。”这命题的真实性是容易用平面几何的方法加以证明的。本文的目的是将这个命题推广到一般。定理:平面上任意位置的两个正 n 边形,其对应顶点连接线的中点是一个 n 边形的 n 个顶点。且中心连接线的中点是这个正 n 边形的中心。
Shanghai High School teacher training textbook “elementary geometry review and research,” Volume 127 on the first series of such a proposition: “Known square ABCD inside a small square EFGH, and M, N, P, Q were AH, BE, CF , The midpoint of DG, then MNPQ is square. ”The truth of this proposition is easily proved by the method of plane geometry. The purpose of this article is to generalize this proposition. Theorem: Two positive n-vertices at any position in the plane, the midpoint of the corresponding vertex connecting line is n vertices of an n-polygon. And the midpoint of the center connecting line is the center of this positive n-polygon.