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Explanation

Given `Deltale(1)/(4)sqrt((a+b+c)abc)`
`rArr(1)/(4Delta)sqrt((a+b+c)abc)ge1`
`rArr ((a+b+c)abc)/(16Delta^(2))ge1`
`rArr (2s abc)/(16Delta^(2))gt1`
`rArr (sabc)/(8*s(s-a)(s-b)(s-c))gt1`
`rArr (abc)/(8s(s-a)(s-b)(s-c))gt1`
`(abc)/(8)ge(s-a)(s-b)(s-c)`
Now, put `s-a=xge 0, s-b=y ge0,s-c=zge0`
`s-a+s-b=x+y`
`2s-a-b=x+y`
`c=x+y`
Similarly, `a=y+z,b=x+z`
`rArr ((x+y))/(2)*((y+z))/(2)*((x+z))/(2)gexyz`
which is true.
Now, equlity will hold if `x=y=z`
`rArr a=b=c`
`rArr` Triangle is equilateral.