Proposition XX. Theorem.
|(98)||Any two sides (A B and A C) of a triangle (B A C) taken together, are greater than the third side (B C).|
Let the side B A be produced, and let A D be cut off equal to A C (III), and let D C be drawn.
Since A D and A C are equal, the angles D and A C D are equal (V). Hence the angle B C D is greater than the angle D, and therefore the side B D in the triangle B C D is greater than B C (XIX). But B D is equal to B A and A C taken together, since A D was assumed equal to A C. Therefore B A and A C taken together are greater than B C.
This proposition is sometimes proved by bisecting the angle A. Let A E bisect it. The angle B E A is greater than E A C, and the angle C E A is greater than E A B (XVI); and since the parts of the angle A are equal, it follows, that each of the angles E is greater than each of the parts of A; and thence, by (XIX), it follows that B A is greater than B E, and A C greater than C E, and therefore that the sum of the former is greater than the sum of the latter.
The proposition might likewise be proved by drawing a perpendicular from the angle A on the side B C; but these methods seem inferior in clearness and brevity to that of Euclid.
Some geometers, among whom may be reckoned Archimedes, ridicule this proposition as being self evident, and contend that it should therefore be one of the axioms. That a truth is considered self evident is, however, not a sufficient reason why it should be adopted as a geometrical axiom (57).
(99) It follows immediately from this proposition, that the difference of any two sides of a triangle is less than the remaining side. For the sides A C and B C taken together are greater than A B; let the side A C be taken from both, and we shall have the side B C greater than the remainder upon taking A C from A B; that is, then the difference between A B and A C.
In this proof we assume something more than is expressed in the fifth axiom. For we take for granted, that if one quantity (a) be greater than another (b), and that equals be taken from both, the remainder of the former (a) will be grater than the remainder of the latter (b). This is a principle which is frequently used, though not directly expressed in the axiom (55).
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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