Question

To provearea of scalen triangle is √s(s-a)(s-b)(s-c)

Answer 1

As Carl pointed out, Heron's formula gives the area of a triangle with sides aa, bb, and ccas A=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√A=s(s−a)(s−b)(s−c), where s=12(a+b+c)s=12(a+b+c).

Let 2s=a+b+c>02s=a+b+c>0, and suppose A2=s(s−a)(s−b)(s−c)A2=s(s−a)(s−b)(s−c) is maximized for aa, bb, and cc that are not all equal. Without loss of generality, assume a<ba<b.

It's straightforward to show that s(s−a+b2)(s−a+b2)(s−c)>A2s(s−a+b2)(s−a+b2)(s−c)>A2, which means that you'll get a triangle of larger area if you replace the unequal sides aa and bb with equal sides, keeping the same perimeter. This contradicts the assumption that a≠ba≠b at the maximum.