If `Delta` is the area of a triangle with side lengths ` a, b, c,` then show that as `Delta leq 1/4 sqrt((a + b + c) abc)` Also, show that the equality occurs in the above inequality if and only if `a = b = c`.

Explanation

We have to prove that
`Delta le (1)/(4) sqrt((a + b + c)abc)`
or `Delta le (1)/(4) sqrt(2 s abc)`
or `Delta^(2) le (1)/(16) 2s abc`
or `Delta^(2) le (1)/(16) 2s Delta 4R`
or `rs le (1)/(2) sR`
Hence, `R ge 2R` [which is always true in `Delta_(2)`]
Alternative Method:
In triangle, sum of two sides is greater than the third side
So `a + b gt c, b + c gt a and c + a gt b`
Now consider quantities `a + b -c, b + c -a, c + a -b`
Using A.M. `ge` G.M. we get
`((a + b -c) + (b + c -a))/(2) ge sqrt((a + b -c) (b + c -a))`
or `b ge sqrt((a + b -c) (b + c -a))`
Similarly we get `c ge sqrt((c +a -b) (b + c -a))`
and `a ge sqrt((a + b -c) (c + a -b))`
Multiplying we get
`abc ge (a + b -c) (b + c -a) (c + a -b)`
`rArr abc ge (2s -2a) (2s -2b) (2s -2c)`
`rArr sabc ge 8s (s -a) (s-b) (s -c)`
`rArr (a + b + c) abc ge 16 Delta^(2)`
`rArr Delta le (1)/(4) sqrt((a + b + c) abc)`