(b+c-a)(c+a-b)(a+b-c)|16(a+b+c)
xyz|16(x+y+z)
xyz同奇偶.
若同奇,xyz|x+y+z,xyz<=x+y+z<=3x,y=z=1或y=3,z=1,于是x=y=z=a=b=c=1,面积不是整数.
于是xyz同偶,记uvw为其一半,则uvw|4(u+v+w).
面积^2=uvw(u+v+w)为平方数.
wlog u>=v>=w
uvw<=4(u+v+w)<=12u,vw<=12,w<=3- w=3时3uv|4(u+v+3)
(3u-4)(3v-4)<=52
3v-4<=7,v=3
9u|4(u+6),无解 - w=2时uv|2(u+v+2)
(u-2)(v-2)<=8
v<=4- v=2时u|4,u=2或4,uvw(u+v+w)不为平方数
- v=3时3u|2(u+5),u=10 (u|2×5,3|u+5)
u,v,w=10,3,2
a,b,c=13,12,5 - v=4时2u|u+6,u=6
u,v,w=6,4,2
a,b,c=10,8,6
- w=1时uv|4(u+v+1),uv(u+v+1)为平方数
(u-4)(v-4)<=20
v<=8- v=8时u=9,a,b,c=17,10,9
- v=7时u<=10,无解
- v=6时3u|2(u+7),u=14,a,b,c=20,15,7
- v=5时5u|4(u+6),u=24,a,b,c=29,25,6
- v=4时u|u+5,u=5,uv(u+v+1)不为平方数
- v=3时3u|4(u+4),u=8,uv(u+v+1)不为平方数
- v=2时u|2(u+3),u=3,a,b,c=5,4,3
- v=1时u|4(u+2),uv(u+v+1)不为平方数
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