上面的代码发现包含一些非法解，最后发现存在bug,修复了两个bug后，得到如附件更多的结果：
其中第六个圆分为半径为$\sqrt{2}$和$\sqrt{5}$俩大类。
比如可以如下面几个图,初始点A和B,其中红色的是最后一步圆和正方形第三个顶点H(但是第二个图H标错了)（6个圆做出正方形第三个顶点），其它各圆的半径为图中同色线段的长度
另外我一直在考虑另外一种做图方案，在做好5个圆以后，前面的方案总是在第6个圆做出一个正方形的顶点，然后第7个圆做出另外一个顶点。
那么有没有可能第7个圆直接做出两个正方形的顶点呢？这个步骤需要第7个圆的圆心在已知边的中垂线上，而且半径长度为这个圆心到正方形另外两个顶点的距离。而这些方案也是现在前面搜索过程唯一可能遗漏的方案。
然后添加搜索方案以后，发现找出了非常多的解，原来对于任何一个前面的方案（第6个圆做出一个正方形的顶点后，由于我们总是已经找出一些点在已知边的中垂线上，那么一这些点为圆心，它们到第6个圆做出的顶点的距离为半径，就可以得到一个同时经过最后两个顶点的圆），但是这些方案显然是平凡的，不是我们需要的。
过滤掉这些方案以后，果然可以发现新的做图方法，比如
如下附件，共可以找到490种不同做图方案（应该有很多类似的做图方案），其中新方案只有8种
mathe 发表于 2021-10-5 13:58
上面的代码发现包含一些非法解，最后发现存在bug,修复了两个bug后，得到如附件更多的结果：
其中第六个 ...
我忘記加一個限制條件了，在Euclidea這個遊戲的4.11出現這道題時，還不能量長度。亦即只能通過現有的點作為圓心，經過另一個點畫圓。
加上限制以后，只找到44个解，但是两种方案都存在，
比如其中第一个方案就是第7个圆同时产生两个顶点，人工挺难构造的。
Find 6th circle center P(1.213525,-0.165396) R0.756934
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Find 6th circle center P(-0.463525,-1.133642) R1.981679
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Circle center P(0.000000,0.000000) R1.000000
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Circle center P(2.000000,0.000000) R1.000000
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Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
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Find 6th circle center P(-0.213525,-0.165396) R0.756934
7th circles center (0.500000,0.866025), Radius 0.517638 to (0.304091,0.386892)
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.000000
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Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
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Circle center P(1.500000,0.866025) R1.732051
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Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
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Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.000000
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Circle center P(0.000000,0.000000) R1.000000
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Find 6th circle center P(0.000000,-1.732051) R2.236068
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7th circles center (0.500000,-0.866025), Radius 1.931852 to (1.866025,0.500000)
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Find 6th circle center P(0.000000,1.732051) R2.236068
7th circles center (0.500000,-0.866025), Radius 1.931852 to (1.866025,0.500000)
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,0.866025) R1.000000
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Circle center P(1.500000,-0.866025) R2.524338
Find 6th circle center P(0.000000,-1.732051) R2.236068
7th circles center (0.500000,0.866025), Radius 0.517638 to (0.133975,0.500000)
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7th circles center (0.500000,-0.866025), Radius 1.931852 to (1.866025,0.500000)
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Circle center P(0.500000,0.866025) R1.000000
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7th circles center (0.500000,-0.866025), Radius 1.931852 to (1.866025,0.500000)
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Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,0.866025) R1.000000
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7th circles center (0.500000,0.866025), Radius 0.517638 to (0.133975,0.500000)
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.000000
Circle center P(1.500000,-0.866025) R1.732051
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7th circles center (0.500000,-0.866025), Radius 1.931852 to (-0.866025,0.500000)
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,0.866025) R1.000000
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7th circles center (0.500000,0.866025), Radius 0.517638 to (0.866025,0.500000)
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.000000
Circle center P(-0.500000,-0.866025) R1.732051
Circle center P(1.228714,-0.973494) R1.732051
Find 6th circle center P(1.000000,-1.732051) R2.236068
7th circles center (0.500000,0.866025), Radius 0.517638 to (0.866025,0.500000)
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.000000
Circle center P(-0.500000,-0.866025) R1.732051
Circle center P(0.271286,0.684819) R1.732051
Find 6th circle center P(1.000000,1.732051) R2.236068
7th circles center (0.500000,-0.866025), Radius 1.931852 to (-0.866025,0.500000)
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,0.866025) R1.000000
Circle center P(-0.500000,0.866025) R1.732051
Circle center P(0.271286,-0.684819) R1.732051
Find 6th circle center P(2.000000,0.000000) R2.236068
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,0.866025) R1.732051
Circle center P(0.000000,0.000000) R2.000000
Circle center P(-1.000000,1.732051) R2.645751
Find 6th circle center P(2.000000,0.000000) R2.236068
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,0.866025) R1.732051
Circle center P(0.000000,0.000000) R2.000000
Circle center P(-1.000000,1.732051) R2.645751
Find 6th circle center P(2.000000,0.000000) R2.236068
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,0.866025) R1.732051
Circle center P(-1.000000,0.000000) R1.732051
Circle center P(-1.000000,1.732051) R2.645751
Find 6th circle center P(2.000000,0.000000) R2.236068
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,0.866025) R1.732051
Circle center P(-1.000000,0.000000) R1.732051
Circle center P(-1.000000,1.732051) R2.645751
Find 6th circle center P(2.000000,0.000000) R2.236068
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.732051
Circle center P(0.000000,0.000000) R2.000000
Circle center P(-1.000000,-1.732051) R2.645751
Find 6th circle center P(2.000000,0.000000) R2.236068
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.732051
Circle center P(0.000000,0.000000) R2.000000
Circle center P(-1.000000,-1.732051) R2.645751
Find 6th circle center P(2.000000,0.000000) R2.236068
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.732051
Circle center P(-1.000000,0.000000) R1.732051
Circle center P(-1.000000,-1.732051) R2.645751
Find 6th circle center P(2.000000,0.000000) R2.236068
Circle center P(0.000000,0.000000) R1.000000
Circle center P(1.000000,0.000000) R1.000000
Circle center P(0.500000,-0.866025) R1.732051
Circle center P(-1.000000,0.000000) R1.732051
Circle center P(-1.000000,-1.732051) R2.645751
结果做成svg格式的html文件在附件中，可以直接用浏览器打开
比如如下的两个图，所有点按照被使用到的顺序予以字母标号
比如下面第一个图中先做了虚线圆A和圆B,两者都过C点。然后做圆D,也过C点。
然后做圆E过F点 (E为圆A和圆D交点，F为圆C和圆D交点).
然后做圆C过A点。
最后做第六圆圆G过H点(G为圆B和圆D交点， H为圆E和圆C交点)
红色圆G和圆A交点I就是所求正方形第三个顶点。最后可以通过第7圆做出正方形另外一个顶点（但是图中没有给出)
图上面存在一点误差，是由于做图过程中四舍五入引起，不是因为计算的原因。原始数据可以看14#
如上图，对结果做了如下更新:
i)坐标使用浮点数，结果图片可以发现更加精确，不会出现看上去圆不经过点的情况
ii)每个图下面按顺序加上了各个圆的圆心和对应的圆上一点，比如第一个总是A(B)代表以A为圆心，过B点做的圆。
可以看到上面复制下来的两个图几乎一模一样，咋一看以为重复了，仔细查看才发现标的G点不同，可以有两个不同的选择。当然两种方案没有本质的区别
可以对于做图后得到的前五个圆相同的方案看成一样的，那么可以减少到只有34种
mathe 发表于 2021-10-7 16:15
可以对于做图后得到的前五个圆相同的方案看成一样的，那么可以减少到只有34种
這個是正解的圖，左右對稱，非常美妙。好像我在你的4.2html裏面沒有見到？
知道4.html里面为什么没有上面的结果了，是因为这几个圆搜索结果如下:
Circle (0,0), 1.0
Circle (1,0), 1.0
Circle(0.5,0.866025), 1.732051
Circle, (-1,0) 2.0
Dep (0,1,3,25)
...
Circle, (2,0) 2.0
Dep (0,1,3,47)
Dep (0, 1, 5,47)
...
而后面继续搜索时，我偷懒直接一个圆的的依赖集出发搜索，而由于我们这里圆25和圆47不会相互依赖，所以就找不到同时使用它们俩的解了。
下面附件简单添加了双圆依赖，从34个增加到40个，其中5,6个就是你上面的方案
还有第三个解也超级对称:
引入递归机制，支持多圆，又添加两个结果，但是仔细比较，发现增加的是重复的解（还没有找原因），所以上面40个解应该全了。