Proposition VI. Theorem.
|(65)||If two angles (B and C) of a triangle (B A C) be equal, the sides (A C and A B) opposed to them are also equal.|
For if the sides be not equal, let one of them A B be greater than the other, and from it cut off D B equal to A C (III), and draw C D.
Then in the triangles D B C and A C B, the sides D B and B C are equal to the sides A C and C B respectively, and the angles D B C and A C B are also equal; therefore (IV) the triangles themselves D B C and A C B are equal, a part equal to the whole, which is absurd; therefore neither of the sides A B or A C is greater than the other; there are therefore equal to one another.
|(66)||Cor.—Hence every equiangular triangle is also equilateral, for the sides opposed to every two equal angles are equal.|
In the construction for this proposition it is necessary that the part of the greater side which is cut off equal to the less, should be measured upon the greater side B A from vertex (B) of the equal angle, for otherwise the fourth proposition could not be applied to prove the equality of the part with the whole.
It may be observed generally, then when a part of one line is cut off equal to another, it should be distinctly specified from which extremity the part is to be cut.
This proposition is what is called by logicians the converse of the fifth. It cannot however be inferred from it by the logical operation called conversion; because, by the established principles of Aristotelian logic, an universal affirmative admits no simple converse. This observation applies generally to those propositions in the Elements which are converses of preceding ones.
The demonstration of the sixth is the first instance of indirect proof which occurs in the Elements. The force of this species of demonstration consists in showing that a principle is true, because some manifest absurdity would follow from supposing it to be false.
This kind of proof is considered inferior to direct demonstration, because it only proves that a thing must be so, but fails in showing why it must be so; whereas direct proof not only shows that the thing is so, but why it is so. Consequently, indirect demonstration is never used, except where no direct proof can be had. It is used generally in proving principles which are nearly self-evident, and in the Elements if oftenest used in establishing the converse propositions. Examples will be seen in the 14th, 19th, 25th and 40th propositions of this book.
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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