X + y + z equals 1 implies xy + yz + zx less than Half/Corollary

Theorem

Let $d$ be a non-zero real number.

Let $x$, $y$ and $z$ be real numbers with:

$x + y + z = d$

Then:

$x y + y z + z x < \dfrac {d^2} 2$

Proof

We have:

$\ds \frac x d + \frac y d + \frac z d = 1$

so, by $x + y + z = 1$ implies $x y + y z + z x < 1/2$, we have:

$\ds \frac x d \times \frac y d + \frac y d \times \frac z d + \frac z d \times \frac x d < 1$

That is:

$\ds \frac 1 {d^2} \paren {x y + y z + z x} < \frac 1 2$

giving:

$\ds x y + y z + z x < \frac {d^2} 2$

$\blacksquare$