either equal or unequal. If they are equal, then since the triangles have two angles of the one equal to two angles of the other, each to each, they are equiangular to one another. We have therefore only to consider the case in which the angles included by the proportional sides are unequal.
Let the triangles ABC, DEF have the angle at A equal to the angle at D, and AB to BC as DE is to EF, but the angle ABC not equal to the angle DEF: the angles ACB and DFE shall be together equal to two right angles.
For, one of the angles ABC, DEF must be greater than the other; suppose ABC the greater; and make the angle ABG equal to the angle DEF. Then it may be shewn, as in VI. 7, that BG is equal to BC, and the angle BGA equal to the angle EFD. Therefore the angles ACB and DFE are together equal to the angles BGC and AGB, that is, to two right angles.
Then the results enunciated in VI. 7 will readily follow. For if the angles ACB and DFE are both greater than a right angle, or both less than a right angle, or if one of them be a right angle, they must be equal.
VI. 8. In the demonstration of VI. 8, as given by Simson, it is inferred that two triangles which are similar to a third triangle are similar to each other; this is a particular case of VI. 2 1, which the student should consult, in order to see the validity of the inference.
VI. 9. The word part is here used in the restricted sense of the first definition of the fifth Book. VI. 9 is a particular case of VI. 10.
VI. 10. The most important case of this proposition is that in which a straight line is to be divided either internally or externally into two parts which shall be in a given, ratio.
The case in which the straight line is to be divided internally is given in the text; suppose, for example, that the given ratio is that of AE to EC; then AB is divided at G in the given ratio.
Suppose, however, that AB is to be divided externally in a given ratio; that is, suppose that AB is to be produced so that the whole straight line made up of AB and the part produced may be to the part produced in a given ratio. Let the given ratio
