两个三角形，有一个角是相同的，它们的边长，是六个不同的一位整数，求这两个三角形

luyuanhong

两个三角形，有一个角是相同的，它们的边长，是六个不同的一位整数，求这两个三角形。

大傻8888888

边长为1、3、4和边长为2、6、8的两个三角形的三个角是都是相同的。

luyuanhong

但是 1+3=4  , 2+6=8 ，不符合“两边之和大于第三边”的要求，不能算是三角形的边。

ataorj

一个思路:
最大边为9的三角形其余两边:
8和2,3,...,7
7和3,4,...,6
6和4,5
-----------
8的:
7和2,3,4,5,6
6和3,4,5
5和4
-----------
7的:
6和2,3,4,5
5和3,4
-----------
6的:
5和2,3,4
4和3
-----------
5的:
4和2,3
-----------
4的:
3和2
========以上排除了1
因子:
2:8=2*2*2,4=2*2
3:9=3*3
2,3:6=2*3,

#=COS∠B=COS∠F
a^2+c^2-2ac#=b^2
#=(a^2+c^2-b^2)/2ac
d^2+e^2-2de#=f^2
#=(d^2+e^2-f^2)/2de

(a^2+c^2-b^2)/ac=(d^2+e^2-f^2)/de
de(a^2+c^2-b^2)=ac(d^2+e^2-f^2)

然后通过上式分析a,b,c有无公共因子...
......

大傻8888888

只有边长为2、3、4的三角形中与3相对角的cos值和边长为6、7、8的三角形中与6相对角的cos值都是11/16符合本题要求.

luyuanhong

谢谢楼上 大傻8888888 的解答。我已将此帖转贴到“陆老师的《数学中国》园地”。

ccmmjj

找到网友，使用计算软件，算出边长在20以内满足一角相等但不相似的两整边三角形如下之多。呵呵
{{2,3,4,7,6,8},{3,8,10,5,4,6},{2,4,5,9,8,10},{2,5,6,4,7,10},{3,10,11,4,5,7},{2,5,6,11,10,12},{2,3,4,8,5,12},{2,11,12,4,5,6},{2,8,9,3,10,12},{5,12,13,6,8,10},{7,8,10,13,12,14},{7,8,12,10,9,14},{7,11,12,9,13,14},{6,12,14,7,8,9},{6,10,14,7,8,13},{6,10,14,7,5,8},{5,6,7,10,8,14},{4,5,7,12,10,14},{4,11,14,10,7,12},{4,11,12,8,13,14},{4,6,7,8,11,14},{4,12,14,7,8,10},{4,7,9,6,12,14},{3,6,8,12,9,14},{3,13,14,7,11,12},{3,7,8,6,10,14},{2,6,7,13,12,14},{2,5,6,10,11,14},{2,13,14,7,10,12},{8,9,15,10,6,12},{7,12,15,9,11,14},{6,8,10,14,13,15},{6,12,15,9,8,10},{5,6,7,11,14,15},{4,5,7,14,11,15},{4,6,9,12,8,15},{4,13,15,6,8,10},{4,9,10,6,14,15},{3,4,5,14,13,15},{3,5,6,12,13,15},{3,5,6,9,8,15},{3,9,10,5,12,15},{2,6,7,4,14,15},{9,11,16,10,8,12},{8,9,10,13,15,16},{8,10,12,11,15,16},{8,13,15,10,14,16},{7,11,16,12,8,14},{7,13,15,10,14,16},{7,12,13,10,14,16},{7,8,12,9,14,16},{6,14,16,8,13,15},{6,14,16,7,13,15},{6,14,16,7,10,13},{5,15,16,10,12,11},{5,6,8,10,13,16},{5,13,16,8,11,15},{5,14,16,6,9,12},{5,14,16,6,7,8},{5,7,8,6,14,16},{4,8,10,15,13,16},{4,6,8,14,12,16},{4,5,6,11,15,16},{4,5,6,11,9,16},{4,5,8,10,9,16},{4,12,14,7,13,16},{4,10,12,5,15,16},{4,7,10,5,15,16},{4,6,8,5,14,16},{3,8,10,15,11,16},{3,5,7,14,10,16},{3,7,8,10,14,16},{3,8,10,9,11,16},{3,14,16,8,7,10},{3,8,10,6,11,16},{3,5,7,6,14,16},{3,8,10,4,15,16},{2,7,8,15,14,16},{2,4,5,15,13,16},{2,3,4,14,12,16},{2,3,4,5,14,16},{2,9,10,3,15,16},{6,7,8,11,16,17},{5,11,12,15,17,16},{5,12,13,8,15,17},{4,15,17,5,8,11},{4,14,17,5,8,12},{3,4,5,8,15,17},{3,16,17,6,9,12},{3,16,17,4,6,8},{10,17,18,12,14,13},{10,14,18,9,15,16},{9,11,12,15,17,18},{9,11,16,15,12,18},{9,10,13,11,15,18},{8,13,14,16,18,17},{8,11,18,15,6,16},{8,13,18,6,14,15},{7,9,12,16,14,18},{7,13,18,14,9,15},{7,17,18,9,11,10},{7,9,12,8,14,18},{6,9,10,14,17,18},{6,10,11,12,15,18},{6,15,18,11,10,12},{6,10,12,11,15,18},{6,15,18,10,11,14},{6,13,14,10,17,18},{6,11,15,10,12,18},{6,16,18,9,13,17},{6,16,18,9,11,12},{6,16,18,8,9,13},{6,9,10,8,16,18},{5,9,10,11,15,18},{5,8,9,10,14,18},{5,7,9,10,16,18},{5,17,18,9,11,10},{5,9,12,7,15,18},{5,9,11,6,16,18},{4,11,12,16,18,17},{4,7,9,16,14,18},{4,8,9,14,17,18},{4,14,15,10,17,18},{4,9,11,10,12,18},{4,9,10,8,13,18},{4,7,10,6,15,18},{3,8,9,15,17,18},{3,8,10,15,12,18},{3,6,7,14,16,18},{3,5,6,11,15,18},{3,17,18,9,14,15},{3,6,7,8,14,18},{3,4,6,8,16,18},{3,10,12,4,16,18},{2,8,9,17,16,18},{9,14,19,10,12,18},{8,12,13,14,16,19},{7,12,13,14,15,19},{7,10,13,11,14,19},{6,7,8,15,16,19},{6,9,12,7,16,19},{5,7,8,14,15,19},{5,16,19,7,8,13},{5,18,19,6,10,12},{5,16,19,6,10,14},{4,10,11,12,14,19},{4,8,9,10,16,19},{4,6,8,7,16,19},{3,7,8,11,14,19},{2,3,4,7,16,19},{2,18,19,6,8,12},{2,18,19,3,4,6}}