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Someyama, Norihiro; Borongan, Mark Lyndon Adamas

New proofs of triangle inequalities. (English) Zbl 1454.51015

J. Ramanujan Soc. Math. Math. Sci. 7, No. 1, 85-92 (2019).
Summary: 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.

MSC:

51M16 Inequalities and extremum problems in real or complex geometry

Keywords:

triangle inequality; circle; ellipse; foci of ellipse; Euclidean geometry; algebraic geometry
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References:

[1] Burk, F., Lebesgue Measure and Integration, WILEY INTERSCIENCE (1998). · Zbl 0886.28001
[2] Dillon, M. I., Geometry Through History, Springer (2018). · Zbl 1395.51002
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