The triangle inequality is basic for many results in real and complex analysis. The geometric form states that the sum of any two sides of a triangle is greater than the third. This was included as Proposition XX in the first book of Euclid's Elements. Many geometric triangle inequalities involving sides, angles, altitudes, inscribed circles and circumscribed circles have been found. Hundreds of these inequalities are summarized in [l] and . A nice geometric proof of the triangle inequality is given in .
Bailey, Herb, "A Stronger Triangle Inequality" (1996). Mathematical Sciences Technical Reports (MSTR). 119.