Chapter 4: Geometry

4.4.4 CONCURRENCE AND COLLINEARITY
Three lines ¼ Ü · ¼ Ý · ¼ ¼, ½ Ü · ½ Ý · ½ ¼, and ¾ Ü · ¾ Ý · ¾ ¼are
concurrent (i.e., intersect at a single point) if and only if
¬ ¼ ¼ ¼¬¬¬¬¬
½ ½ ½ ¼ (4.4.11)
¬ ¾ ¾ ¾
(This remains true in oblique coordinates.)
Three points ´Ü ¼ Ý ¼ µ, ´Ü ½ Ý ½ µ, and ´Ü ¾ Ý ¾ µ are collinear (i.e., all three points
are on a straight line) if and only if
¬
¬
Ü ¼ Ý ¼ ½
Ü ½ Ý ½ ½ ¼ (4.4.12)
Ü ¾ Ý ¾ ½
(This remains true in oblique coordinates.)
Three points with polar coordinates ´Ö ¼ ¼ µ, ´Ö ½ ½ µ, and ´Ö ¾ ¾ µ are collinear
if and only if
Ö ½ Ö ¾ ×Ò´ ¾ ½ µ·Ö ¼ Ö ½ ×Ò´ ½ ¼ µ·Ö ¾ Ö ¼ ×Ò´ ¼ ¾ µ¼ (4.4.13)
4.5 POLYGONS
Given ¿ points ½ in the plane, in a certain order, we obtain a -sided
polygon or -gon by connecting each point to the next, and the last to the rst, with a
line segment. The points are the vertices and the segments ·½ are the sides
or edges of the polygon. When ¿we have a triangle, when we have a
quadrangle or quadrilateral, and so on (see page 324 for names of regular polygons).
Here we will assume that all polygons are simple: this means that no consecutive
edges are on the same line and no two edges intersect (except that consecutive edges
intersect at the common vertex) (see Figure 4.9).
FIGURE 4.9
Two simple quadrilaterals (left and middle) and one that is not simple (right). We will treat
only simple polygons.
¾
¿
½
¿
½
¾
¿
½
¾
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