`a;b;c` nguyên dương
`=>a;b;c>0`
Vì `a/b={2b}/c=>2b^2=ac`
`\qquad a/b={4c}/a=>a^2=4bc`
`\qquad {2b}/c={4c}/a=>4c^2=2ab`
`=>2c^2=ab`
`=>{2b^2}/{2c^2}={ac}/{ab}`
`=>{b^2}/{c^2}=c/b`
`=>b^3=c^3=>b=c`
$\\$
`\qquad a^2=4bc`
`=>a^2=4b.b=(2b)^2`
`=>a=2b` (vì $a;b>0$)
Thay `a=2b;c=b` vào biểu thức $T$
`T={ab+bc+ca}/{a^2+b^2+c^2}`
`T={2b.b+b.b+b.2b}/{(2b)^2+b^2+b^2}`
`T={2b^2+b^2+2b^2}/{4b^2+b^2+b^2}`
`T={5b^2}/{6b^2}=5/ 6`
Vậy `T=5/ 6`