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-
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.
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.