Motivated by Napoleon theorem, we study the properties of the triangles obtained by moving the midpoint of each side of a given trianle along the perpendicular bisector of corresponding sides, and extend the results to the case of quadrilaterals. On the other hand ,we consider the method of erecting a regular M-gon to each side of a given N-gon and joint the N centers of these M-gons to form a new N-gon. (abbreviated as CRG method),and get the following results. 1. We characterize some kinds of N-gons that can be transformed to regular N-gons via CRG method. 2. Of M,N are nature numbers with M|N, then it is possible to find a N-gon that can be transformed to a regular N-gon by CRG method. \r 3. If a polygon P is symmetric with respect to a fixed point or a fixed line, then P can be transformed by CRG to a polygon with similar symmetries. 4. If a polygon P is transformed by CRG to ′P,there exists a commonpoint G such that ΣGA=0 andΣGB=0, where A and B runs through vertices of and P′P, respectively. 本研究將拿破崙定理加以延伸。先探討由各邊中點沿中垂線延伸得出之三角形的性質並推廣至四邊形之情形條列式報告成果。另一個推廣是將給定的多邊形的每邊外接一個正多邊形,再以這些外接的正多邊形的中心為頂點造出一個新的多邊形。我們發現此幾何變換具有以下性質:(1) 「哪些多邊形能被變換成正多邊形呢?」,我們觀察出能被變換成正多邊形的多邊形其限制條件隨邊數增加而增多,並進一步區分了哪些多邊形可以被變換成正多邊形。 (2) 在將非正N邊形做變換時,不一定須外接正N邊形才能得到正N邊形,我們區分出可外接哪些正多邊形而得到正多邊形。 (3) 對給定的多邊形作此變換時,若原多邊形有點對稱或線對稱等性質,則新多邊形也將具有相同的性質。 (4) 此變換得到的新多邊形會與原多邊形共重心,亦即新舊兩多邊形內到各自的頂點向量和為0的點會是同一點。
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