PROPORTIONS OF THE TRIANGLES. 89
In fact, changing nothing of the triangle abc, let us make it turn,tothe extent of a right angle, round the point a, then ac will placeitsel in ac' in a position parallel to AC ; it will be the same for ah',and b' c'; whence the triangle a'b'c' will have its sides parallel to thoseof ABC, and the two triangles will be similar. Consequently ABCand abc are also similar.
When the sides of two triangles are proportional, theircorresponding angles are equal, and the triangles are si-milar. Let us suppose that the two triangles ABC a'b'c?fig. 7, pi. 5, have no other relation than this.
AB : a’b' : : AC : a'c' :: BC : b'c'.
Let us imagine a second triangle abc having the sideab = a’b', and its three sides parallel, respectively to AB,BC, and AC , we shall then have A : ab : : AC : ac BC :be. Whence
b’c? =
AB
BC
AB
If ah’ = ab, a'c? must be equal to ac and b'c' = be.
The two triangles abc, a'b’c', therefore, have their threesides respectively equal and they are consequently equal:thus the angles a — a = A, b' = b = B, d = c — C.
Thus, whenever the sides of two triangles are propor-tionals, the angles opposite the proportional sides are equal,and the triangles are similar.
When two triangles, ABC, abc, have the sides AB,AC , proportional to ab, ac, and the angle A = fl, the twotriangles are similar ; for if we place the angle a on A, theproportional AB : ab : : AC : ac requires that AC and acshould be parallel; then all the three sides are parallel.
If we draw, from the point O, fig. 6, pi. 5, three rightlines OPR, OQS, OTU, cutting the two parallel linesPTQ, RUS, we shall have successively, in consequenceof the similar triangles,
OPT, ORU; 1st. . . . OT : OU : : PT : RU,OQT, OSU; 2d. . . . OT : OU : : QT : SU,and finally, PT : RU : : QT : SU-