1sin a/2 sin b/2 sin c/2 = (s -a ) (s - b ) ( s- c ) / abc { by half angle formulas }
so given ( s - a ) ( s - b ) ( s - c ) / abc ≤ 1 / 8
or rearrange to get ( 2s - 2a ) ( 2s - 2b ) ( 2s - 2c ) ≤ abc
or ( a + b - c ) ( a + b - c ) ( b + c -a ) ≤ abc -------------------------------------------- 1
this is what we have to prove --
now a2 ≥ a2 - ( b - c )2 = ( a + b - c ) ( a - b + c )
similarly get two more equations , and then multiply to get
a2 b2 c2 ≥ ( a + b - c )2 ( a + c - b )2 ( b + c - a )2 -------------------------------------- 2
combining 1 and 2 ,
you get the result .
