The sides of a triangle are \(a\), \(b\), \(c\), and \(s\) is given by \(2s = a + b + c\).

Prove that

  1. \(4(s-b)(s-c) \le a^2\),
  2. \(8(s-a)(s-b)(s-c) \le abc\).

State under what conditions the sign of equality applies, in each case.

Investigate the truth of the results (i) and (ii) if \(a\), \(b\), \(c\) are any positive quantities, not necessarily the sides of a triangle.