Editing Triangle inequality (section)

===Mathematical expression of the constraint on the sides of a triangle===

For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths {{math|''a'', ''b'', ''c''}} that are all positive and excludes the degenerate case of zero area):
:<math>a + b > c ,\quad b + c > a ,\quad c + a > b .</math>
A more succinct form of this inequality system can be shown to be
:<math>|a - b| < c < a + b .</math>
Another way to state it is
:<math>\max(a, b, c) < a + b + c - \max(a, b, c)</math>
implying
:<math>2 \max(a, b, c) < a + b + c</math>
and thus that the longest side length is less than the [[semiperimeter]].

A mathematically equivalent formulation is that the area of a triangle with sides {{math|''a'', ''b'', ''c''}} must be a real number greater than zero. [[Heron's formula]] for the area is

:<math>
\begin{align}
4\cdot \text{area} & =\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\
& = \sqrt{-a^4-b^4-c^4+2a^2b^2+2a^2c^2+2b^2c^2}.
\end{align}
</math>

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).

The triangle inequality provides two more interesting constraints for triangles whose sides are {{math|''a'', ''b'', ''c''}}, where {{math|''a'' ≥ ''b'' ≥ ''c''}} and <math>\phi</math> is the [[golden ratio]], as
:<math>1<\frac{a+c}{b}<3</math>

:<math>1\le\min\left(\frac{a}{b}, \frac{b}{c}\right)<\phi.</math><ref>''[[American Mathematical Monthly]]'', pp. 49-50, 1954.</ref>