A property of sides of a triangle?

Prove or disprove:

Proposition: $a, b, c$ are sides of a triangle if

$$ (a+b+c)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \lt 10$$

Note that we can easily show that (using AM $\ge$ GM for instance)

$$ (a+b+c)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \ge 9$$

for any positive reals $a, b, c$.

So we can restate the above statement as

Proposition: $a, b, c$ are sides of a triangle if

$$\left\lfloor (a+b+c)\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \right\rfloor = 9$$

where $\lfloor x \rfloor$ is the integer part of $x$.


[Note: Initially I had if only if, but I have changed the statements to just be an implication]