本文发表于《中学生数学》（高中版）2008年第8期（上半月）
正七边形的一个特殊性质
215006 苏州市第一中学 刘祖希
正七边形是一个十分特殊的多边形，它的一个显著特征是蕴含内角为![](https://wkrtcs.bdimg.com/rtcs/image?w=73&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"73","h":"41","dataType":"gif","c":"word/media/image1.png"})
（或比值为）的三角形，这一点深受数学竞赛的青睐.
例1 ![](https://wkrtcs.bdimg.com/rtcs/image?w=47&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"47","h":"19","dataType":"gif","c":"word/media/image3.png"})
的边长为，对应角的大小之比为，求证：.
这是一个经典的问题，下面这道题完整地回答了它.
例2 （1987年全苏数学奥林匹克）设![](https://wkrtcs.bdimg.com/rtcs/image?w=80&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"80","h":"24","dataType":"gif","c":"word/media/image7.png"})
是圆内接正七边形，求证：
![](https://wkrtcs.bdimg.com/rtcs/image?w=131&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"131","h":"45","dataType":"gif","c":"word/media/image8.png"})
.
证明：如图，各顶点之间的距离只有三种可能值，由小到大依次记为![](https://wkrtcs.bdimg.com/rtcs/image?w=41&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"41","h":"17","dataType":"gif","c":"word/media/image9.png"})
，
![](https://wkrtcs.bdimg.com/rtcs/image?w=150.4&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"150.4","h":"154.4","dataType":"png","c":"word/media/image10.png","imgOriW":188,"imgOriH":193})
在四边形中，由托勒密定理，
![](https://wkrtcs.bdimg.com/rtcs/image?w=237&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"237","h":"24","dataType":"gif","c":"word/media/image12.png"})
，
即![](https://wkrtcs.bdimg.com/rtcs/image?w=83&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"83","h":"19","dataType":"gif","c":"word/media/image13.png"})
，，
等价于.
例3 （2003年江苏夏令营试题）将平面上的点都以红、蓝两色之一着色，证明：存在有两个内角分别是![](https://wkrtcs.bdimg.com/rtcs/image?w=55&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"55","h":"41","dataType":"gif","c":"word/media/image15.png"})
且它们的夹边长为2003的三角形，其三个顶点同色.
证明：如图，考察一个边长为2003的圆内接正七边形.
由抽屉原理，必有四个顶点同色，且此四点必有两点相邻，
不妨设同蓝色，
①![](https://wkrtcs.bdimg.com/rtcs/image?w=40&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"40","h":"24","dataType":"gif","c":"word/media/image17.png"})
有一蓝色，则或满足要求；
②均为红色，且有一红色，
则![](https://wkrtcs.bdimg.com/rtcs/image?w=56&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"56","h":"24","dataType":"gif","c":"word/media/image21.png"})
或满足要求；
③均为红色，且均为蓝色，
则![](https://wkrtcs.bdimg.com/rtcs/image?w=56&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"56","h":"24","dataType":"gif","c":"word/media/image23.png"})
和满足要求.
命题得证.
例4 （1999年世界城市际赛题）已知![](https://wkrtcs.bdimg.com/rtcs/image?w=13&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"13","h":"16","dataType":"gif","c":"word/media/image25.png"})
为内心，连.若中有一个三角形与相似（我们称它们为母子三角形），求各角的大小.
![](https://wkrtcs.bdimg.com/rtcs/image?w=146&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"146","h":"150","dataType":"png","c":"word/media/image28.png","imgOriW":182,"imgOriH":188})
解：不妨设与相似，则，
否则，若![](https://wkrtcs.bdimg.com/rtcs/image?w=81&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"81","h":"19","dataType":"gif","c":"word/media/image31.png"})
，即，，矛盾.
不失一般性，设![](https://wkrtcs.bdimg.com/rtcs/image?w=248&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"248","h":"20","dataType":"gif","c":"word/media/image34.png"})
，
则![](https://wkrtcs.bdimg.com/rtcs/image?w=274&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"274","h":"41","dataType":"gif","c":"word/media/image35.png"})
，
解得![](https://wkrtcs.bdimg.com/rtcs/image?w=189&md5sum=57721629c39ac6212406806862db64fd&sign=c7c94d6972&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=75&ipr={"t":"img","w":"189","h":"41","dataType":"gif","c":"word/media/image36.png"})
，
因此，各角分别为.
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