线段AC上有一点B。以AB和BC为边，分别构造等边三角形XAB和YBC。证明，以下七个点共圆：点B，XC和YA的交点，以及线段AC、XC、YA、XB、YB的中点。
如图，S为AB的中点，T为BC的中点。首先注意到，△ABY和△XBC全等(SAS)，也就是说△XBC是△ABY绕B点旋转60°得到的，因此AY和XC的夹角∠AMX=60°。注意到中位线LQ∥AX，中位线KQ∥CY，我们立即得到∠KQL也等于60°，于是K、L、M、Q四点共圆。
另外，中位线NL∥BC，中位线KP∥AB，因此线段NL、KP和AC全都平行。又中位线PT∥YC，YC∥XB，而XB与中位线LT平行，因此PT∥LT，即P、L、T三点共线。于是我们得知∠LPK=∠LTB=60°。类似地，∠NKP也等于60°，因此四边形KNLP是一个等腰梯形，这四个点共圆。而∠LPK也是弦KL所对的圆周角，因此这个圆和前面的那个圆是同一个圆，L、N、K、Q、P、M都在这个圆上。又∠NBP=60°=∠NKP，它们都是NP所对的圆周角，因此B也在这个圆上。
来源：http://www.cut-the-knot.org/Curriculum/Geometry/TwoEquilateralBumps.shtml
a sofa
先占个座位
翻译得不错 似乎那个网站很棒啊 看原文更有感觉 呵呵
看见SAS总是让我想起初中时无穷无尽的几何证明题
不错，感觉上这个很有意思。
我记得……狗狗给我说过个类似的……
=.=好像叫做九点共圆吧……
九点共圆~~诡异的东东~~MS和三角形滴几条线有关~
还有个“五点共圆”：
http://www.cyol.com/zqb/content/2001-04/06/content_200107.htm
《中国青年报》有个“五点共圆”问题：
http://www.cyol.com/zqb/content/2001-04/06/content_200107.htm
中国青年报 China Youth Daily 2001年4月6日 星期五
为什么网址贴不上来：
h t t p : / / w w w . c y o l . c o m / z q b / c o n t e n t / 2 0 0 1 – 0 4 / 0 6 / c o n t e n t _ 2 0 0 1 0 7 . h t m
看了那么多字母就晕了……
虽然能看懂……
NL∥AC∥KP , KN∥AX∥LQ∥PB , LP∥XB
NK=0.5(AX-BY)=0.5(XB-YC)=LP => NKPL等腰梯形 => NKPL共圆
NL=0.5BC , KQ=0.5YC , BP=0.5BY => NKQL等腰梯形，NBPL等腰梯形 => NKQL共圆，NBPL共圆
△YAB≌△CXB => ∠YAB=∠CXB => ∠LMK=∠XBA=∠KPL => KPML共圆
故NKQBPML共圆
NL∥AC∥KP , KN∥AX∥LQ∥PB , LP∥XB
NK=0.5(AX-BY)=0.5(XB-YC)=LP => NKPL等腰梯形 => NKPL共圆
NL=0.5BC , KQ=0.5YC , BP=0.5BY => NKQL等腰梯形，NBPL等腰梯形 => NKQL共圆，NBPL共圆
△YAB≌△CXB => ∠YAB=∠CXB => ∠LMK=∠XBA=∠KPL => KPML共圆
故NKQBPML共圆
话说其实不能算美妙，等边三角形和中点群太imba了..
NL//AC//KP , KN//AX//LQ//PB , LP//XB
NK=0.5(AX-BY)=0.5(XB-YC)=LP => NKPL等腰梯形 => NKPL共圆
NL=0.5BC , KQ=0.5YC , BP=0.5BY => NKQL等腰梯形，NBPL等腰梯形 => NKQL共圆，NBPL共圆
△YAB全等△CXB => ∠YAB=∠CXB => ∠LMK=∠XBA=∠KPL => KPML共圆
故NKQBPML共圆
这数学题是什么时候的，记得初中也有类似的吧。
This weeқ the Swedish fսrnishings large IKEA changed a film theater’s seats Mobile customers cⅼick օn right
hеrе to see the Ikea bedroom cinema in motion.
Feel fｒee to surf to my web-site … Accessories For Play Kitchen
I every time spent my half an hour to read this blog’s posts daily along with
a cup of coffee.
Attractive part of content. I simply stumbled upon your web site and in accession capital to assert that I acquire actually enjoyed account your blog posts.
Anyway I’ll be subscribing to your augment and even I achievement you get
admission to constantly quickly.
You actually make it seem so easy with your presentation but I find this
matter to be really something that I think I would never understand.
It seems too complex and extremely broad for me. I’m looking forward
for your next post, I will try to get the hang of it!
That is very fascinating, You’re an excessively skilled blogger.
I have joined your feed and look forward to in quest of extra of your fantastic post.
Also, I’ve shared your web site in my social networks
Howdy! This is kind of off topic but I need
some advice from an established blog. Is it very difficult to set up your
own blog? I’m not very techincal but I can figure things
out pretty quick. I’m thinking about making my own but I’m not
sure where to start. Do you have any ideas or suggestions?
Thanks
I simply couldn’t leave your web site before suggesting that Iextremely enjoyed the usual info an individual supply to your guests?
Is gonhna be back often to check out new posts
my web page … http://www.Toprevenuegate.Com