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Proposition 20. In any triangle two sides taken together in any manner are greater than the remaining one. For let ABC be a triangle; I say that in the triangle ABC two sides taken together in any manner are greater than the remaining one, namely BA, AC greater than BC, AB, BC greater than AC, BC, CA greater than AB.
- Euclid, Elements, BOOK I. - Perseus Digital Library
20. Of trilateral figures, an equilateral triangle is that...
- Euclid, Elements, book 1, type Prop, number 20 - Perseus Digital Library
Euclid, Elements J. L. Heiberg, Ed. ... Click anywhere in...
- Euclid, Elements, BOOK I. - Perseus Digital Library
20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
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This proposition on the triangle inequality, along with I.15 on vertical angles, allows us to solve a problem on minimum distance, described and solved by Heron of Alexandria (ca. 10–ca. 70).
Prop. 20: The two sides of any triangle, taken in any way, are larger than the remaining. (general diagram) (diagram 1) For let there be a triangle, ABG.
Proposition 20 of Book I of Euclid’s Elements of Geometry establishes the Triangle Inequality. This proposition asserts that the sum of any two sides of a triangle is greater than the remaining side. Proclus (as translated by Thomas Taylor, 1792) begins his commentary on this proposition as follows.
Book 1 outlines the fundamental propositions of plane geometry, includ- ing the three cases in which triangles are congruent, various theorems involving parallel lines, the theorem regarding the sum of the angles in a triangle, and the Pythagorean theorem.
