PRINCIPIA · THEOREM

The difference of two sides is less than the third

Dependencies: Triangle inequality a + b > c (the sum of any two sides is greater than the third).

Statement

Let $\triangle ABC$ be any triangle, with sides $a = BC$ , $b = CA$ , $c = AB$ . Then the absolute difference of any two sides is less than the third:

|a - b| < c,\qquad |b - c| < a,\qquad |c - a| < b.

By symmetry it suffices to prove the first; the others follow by relabeling.

The difference of two sides is less than the third: lining up the three lengths, the "short stick" |a - b| is plainly shorter than c

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