We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the triangle inequality as simple as possible without using properties of triangles. In other words, we suggest proofs without using paper and pen.
Keywords and Phrases
Triangle inequality, circle, ellipse, focci of ellipse, Euclidean geometry, algebraic geometry.
A.M.S. subject classiﬁcation
26D05, 26D15, 51M16.