PRINCIPIA · THEOREM

Triangle inequality

Dependencies: Ruler axiom (lay off equal segments on an existing ray), Base angles of an isoceles triangle are equal, Bigger angle ↔ longer opposite side.

Statement

Let △ABC\triangle ABC be any triangle. Then the sum of any two sides is greater than the third:

AB+AC>BC,AB+BC>AC,AC+BC>AB.AB + AC > BC,\qquad AB + BC > AC,\qquad AC + BC > AB.

By symmetry, it suffices to prove the first inequality AB+AC>BCAB + AC > BC; the others follow by relabeling.

Triangle inequality: AB + AC > BC, equivalently |AB - AC| < BC < AB + AC

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