阿波罗尼斯圆及其应用
数学理论
1.“阿波罗尼斯圆”：在平面上给定两点![](https://wkrtcs.bdimg.com/rtcs/image?w=32&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"32","h":"20","dataType":"png","c":"word/media/image1_1.png","imgOriW":846,"imgOriH":529})
，设点在同一平面上且满足当且时，点的轨迹是个圆，称之为阿波罗尼斯圆。
（![](https://wkrtcs.bdimg.com/rtcs/image?w=36&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"36","h":"18.6","dataType":"png","c":"word/media/image6_1.png","imgOriW":952,"imgOriH":493})
时点的轨迹是线段的中垂线）
2.阿波罗尼斯圆的证明及相关性质
![](https://wkrtcs.bdimg.com/rtcs/image?w=243.13&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_8","w":"243.13","h":"161.07","dataType":"png","c":"word/media/image8.png","imgOriW":388,"imgOriH":257})
定理：为两已知点，分别为线段的定比为的内外分点，则以为直径的圆上任意点到两点的距离之比为
证 （以![](https://wkrtcs.bdimg.com/rtcs/image?w=36&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"36","h":"18.6","dataType":"png","c":"word/media/image17_1.png","imgOriW":952,"imgOriH":493})
为例）
设![](https://wkrtcs.bdimg.com/rtcs/image?w=149.33333333333&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"149.33333333333","h":"44","dataType":"png","c":"word/media/image18_1.png","imgOriW":2499,"imgOriH":736})
，则
![](https://wkrtcs.bdimg.com/rtcs/image?w=310.66666666667&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"310.66666666667","h":"41.333333333333","dataType":"png","c":"word/media/image19_1.png","imgOriW":3788,"imgOriH":504})
.
由相交弦定理及勾股定理知
![](https://wkrtcs.bdimg.com/rtcs/image?w=338.66666666667&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"338.66666666667","h":"44","dataType":"png","c":"word/media/image20_1.png","imgOriW":3845,"imgOriH":500})
于是![](https://wkrtcs.bdimg.com/rtcs/image?w=193.33333333333&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"193.33333333333","h":"45.333333333333","dataType":"png","c":"word/media/image21_1.png","imgOriW":2826,"imgOriH":662})
而![](https://wkrtcs.bdimg.com/rtcs/image?w=49.333333333333&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"49.333333333333","h":"21.333333333333","dataType":"png","c":"word/media/image23_1.png","imgOriW":1305,"imgOriH":564})
同时在到两点距离之比等于的曲线（圆）上，不共线的三点所确定的圆是唯一的，因此，圆上任意一点到两点的距离之比恒为
性质1.当![](https://wkrtcs.bdimg.com/rtcs/image?w=36&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"36","h":"18.6","dataType":"png","c":"word/media/image26_1.png","imgOriW":952,"imgOriH":493})
时，点在圆内，点在圆外；
当![](https://wkrtcs.bdimg.com/rtcs/image?w=60&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_7","w":"60","h":"18.6","dataType":"png","c":"word/media/image29_1.png","imgOriW":1587,"imgOriH":493})
时，点在圆内，点在圆外。
性质2.因![](https://wkrtcs.bdimg.com/rtcs/image?w=104&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"104","h":"24","dataType":"png","c":"word/media/image32_1.png","imgOriW":2751,"imgOriH":635})
，过是圆的一条切线。
若已知圆![](https://wkrtcs.bdimg.com/rtcs/image?w=16&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"16","h":"18.6","dataType":"png","c":"word/media/image35_1.png","imgOriW":423,"imgOriH":493})
及圆O外一点，可以作出与之对应的点反之亦然。
性质3.所作出的阿波罗尼斯圆的直径为![](https://wkrtcs.bdimg.com/rtcs/image?w=85.333333333333&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"85.333333333333","h":"49.333333333333","dataType":"png","c":"word/media/image38_1.png","imgOriW":1756,"imgOriH":1015})
，面积为
性质4.过点![](https://wkrtcs.bdimg.com/rtcs/image?w=16&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"16","h":"17.333333333333","dataType":"png","c":"word/media/image40_1.png","imgOriW":423,"imgOriH":458})
作圆的切线为切点），则分别为的内、外角平分线。
性质5.过点![](https://wkrtcs.bdimg.com/rtcs/image?w=16&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"16","h":"16","dataType":"png","c":"word/media/image45_1.png","imgOriW":423,"imgOriH":423})
作圆不与重合的弦则平分
数学应用
1.（03北京春季）设![](https://wkrtcs.bdimg.com/rtcs/image?w=145.33333333333&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"145.33333333333","h":"21.333333333333","dataType":"png","c":"word/media/image51_1.png","imgOriW":3443,"imgOriH":505})
为两定点，动点到点的距离与到点的距离之比为定值求点的轨迹.
2.（05江苏）圆![](https://wkrtcs.bdimg.com/rtcs/image?w=18.6&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"18.6","h":"22.666666666667","dataType":"png","c":"word/media/image57_1.png","imgOriW":493,"imgOriH":599})
和圆的半径都是1，，过动点分别作圆和圆的切线分别为切点），使得，试建立适当坐标系，求动点的轨迹方程.
3.（06四川）已知两定点![](https://wkrtcs.bdimg.com/rtcs/image?w=104&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"104","h":"21.333333333333","dataType":"png","c":"word/media/image64_1.png","imgOriW":2751,"imgOriH":564})
如果动点满足，则点的轨迹所围成的图形的面积是________________.
4.(08江苏)满足条件![](https://wkrtcs.bdimg.com/rtcs/image?w=136&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_3","w":"136","h":"24","dataType":"png","c":"word/media/image68_1.png","imgOriW":3146,"imgOriH":555})
的面积的最大值是___________.
5.在等腰![](https://wkrtcs.bdimg.com/rtcs/image?w=46.666666666667&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_2","w":"46.666666666667","h":"18.6","dataType":"png","c":"word/media/image70_1.png","imgOriW":1234,"imgOriH":493})
中，是腰上的中线，且则面积的最大值是___________.
6.已知![](https://wkrtcs.bdimg.com/rtcs/image?w=72&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_2","w":"72","h":"21.333333333333","dataType":"png","c":"word/media/image74_1.png","imgOriW":1905,"imgOriH":564})
是圆上任意一点，问在平面上是否存在一点，使得若存在，求出点坐标；若不存在，说明理由.
变式：已知圆![](https://wkrtcs.bdimg.com/rtcs/image?w=137.33333333333&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_2","w":"137.33333333333","h":"24","dataType":"png","c":"word/media/image75_1.png","imgOriW":3162,"imgOriH":553})
，问在轴上是否存在点和点，使得对于圆上任意一点，都有若存在，求出坐标；若不存在，说明理由.
7.在![](https://wkrtcs.bdimg.com/rtcs/image?w=46.666666666667&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_2","w":"46.666666666667","h":"18.6","dataType":"png","c":"word/media/image84_1.png","imgOriW":1234,"imgOriH":493})
中，是的平分线，且
（1）求![](https://wkrtcs.bdimg.com/rtcs/image?w=13.333333333333&md5sum=477275e3e1b8eed375ac96ccf6abd502&sign=1b3a30a6e2&rtcs_flag=1&rtcs_ver=4&l=webapp&bucketNum=490&ipr={"t":"img","r":"r_2","w":"13.333333333333","h":"18.6","dataType":"png","c":"word/media/image88_1.png","imgOriW":352,"imgOriH":493})
的取值范围；
（2）若的面积为1，求为何值时，最短.
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