II. ON TRIANGLES 27 II. For any three points A, B, C, each of the distances AB, BC, AB is at most equal to the sum of the other two, and at least equal to the difference of the other two, equality holding if the three points are collinear. Theorem. A line segment is shorter than any broken line with the same end- points. Figure 27 If the broken line has only two sides, the theorem reduces to the preceding one. Consider next a broken line with three sides ABCD (Fig. 27). Drawing BD, we will have AD AB + BD and, since BD BC + CD, we have AD AB + BD AB + BC + CD. The theorem is thus proven for a broken line with three sides. The same reasoning can be used successively for broken lines with 4 sides, 5 sides, etc. Therefore the theorem is true no matter how large the number of sides. 27. The sum of the sides of a polygon, or of a broken line, is called its perimeter. Theorem. The perimeter of a convex broken line is less than that of any broken line with the same endpoints which surrounds it. Figure 28 Consider the convex line ACDB and the surrounding line AC D E F B (Fig. 28). We extend sides AC, CD in the sense indicated by ACDB that is, the side AC past C and the side CD past D. These extensions do not intersect the interior of polygon ACDB because of its convexity. Suppose they cut the surrounding line in G and H respectively.

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