Proposition XXIV. Theorem.
|(103)||If two triangles (E F D, B A C) have two sides of the one respectively equal to two sides of the other (F E to A B and F D to A C), and if one of the angles (B A C) contained by the equal sides be greater than the other (E F D), the side (B C) which is opposite to the greater angle is greater than the side (E D) which is opposite to the less angle.|
In the triangles B A G and E F D the sides B A and A G are equal respectively to E F and F D, and the included angles are equal (const.), and therefore B G is equal to E D. Also, since A G is equal to F D by const., and A C is equal to it by hyp., A G is equal to A C, therefore the triangle G A C is isosceles, and therefore the angles A C G and A G C are equal (V); but the angle B G C is greater than A G C, therefore greater than A C G, and therefore greater than B C G; then in the triangle B G C the angle B G C is greater than B C G, therefore the side B C is greater than B G (XIX), but B G is equal to E D, and therefore B C is greater than E D.
In this demonstration it is assumed by Euclid, that the points A and G will be on different sides of B C, or, in other words, that A H is less than A G or A C. This may be proved thus:—The side A C not being less than A B, the angle A B C cannot be less than the angle A C B (XVIII). But the angle A B C must be less than the angle A H C (XVI); therefore the angle A C B is less than A H C, and therefore A H less than A C or A G (XIX).
In the construction for this proposition Euclid has omitted the words ‘with the side which is not the greater.’ Without these it would not follow that the point G would fall below the base B C, and it would be necessary to give demonstrations for the cases in which the point G falls on, or above the base B C. On the other hand, if these words be inserted, it is necessary in order to give validity to the demonstration, to prove as above, that the point G falls below the base.
If the words ‘with the side not the greater’ be not inserted, the two omitted cases may be proved as follows:
If the point G fall on the base B C, it is evident that B G is less than B C (51).
If G fall above the base B C, let it be at G′. The sum of the lines B G′ and A G′ is less than the sum of A C and C B (XXI). The equals A C and A G′ being taken away, there will remain B G′ less than B C.
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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