Editing Triangle inequality (section)

===Converse===

The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.

In either case, if the side lengths are {{mvar|a}}, {{mvar|b}}, {{mvar|c}} we can attempt to place a triangle in the [[Euclidean plane]] as shown in the diagram. We need to prove that there exists a real number {{mvar|h}} consistent with the values {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}, in which case this triangle exists.

[[Image:Triangle with notations 3.svg|thumb|270px|Triangle with altitude {{mvar|h}} cutting base {{mvar|c}} into {{math|''d'' + (''c'' − ''d'')}}.]]

By the [[Pythagorean theorem]] we have {{math|''b''{{sup|2}} {{=}} ''h''{{sup|2}} + ''d''{{sup|2}}}} and {{math|''a''{{sup|2}} {{=}} ''h''{{sup|2}} + (''c'' − ''d''){{sup|2}}}} according to the figure at the right. Subtracting these yields {{math|''a''{{sup|2}} − ''b''{{sup|2}} {{=}} ''c''{{sup|2}} − 2''cd''}}. This equation allows us to express {{mvar|d}} in terms of the sides of the triangle:
:<math>d=\frac{-a^2+b^2+c^2}{2c}.</math>
For the height of the triangle we have that {{math|''h''{{sup|2}} {{=}} ''b''{{sup|2}} − ''d''{{sup|2}}}}. By replacing {{mvar|d}} with the formula given above, we have

:<math>h^2 = b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2.</math>

For a real number {{mvar|h}} to satisfy this, {{math|''h''{{sup|2}}}} must be non-negative:
:<math>\begin{align}
  0 &\le b^2-\left(\frac{-a^2+b^2+c^2}{2c}\right)^2 \\[4pt]
  0 &\le \left(b- \frac{-a^2+b^2+c^2}{2c}\right) \left(b + \frac{-a^2+b^2+c^2}{2c}\right) \\[4pt]
  0 &\le \left(a^2-(b-c)^2)((b+c)^2-a^2 \right) \\[6pt]
  0 &\le (a+b-c)(a-b+c)(b+c+a)(b+c-a) \\[6pt]
  0 &\le (a+b-c)(a+c-b)(b+c-a)
\end{align}</math>
which holds if the triangle inequality is satisfied for all sides. Therefore, there does exist a real number <math>h</math> consistent with the sides <math>a, b, c</math>, and the triangle exists. If each triangle inequality holds [[strict inequality|strictly]], <math>h > 0</math> and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so <math>h = 0</math>, the triangle is degenerate.