Chapter 4: Geometry

When we refer to the angle at a vertex we have in mind the interior angle
(as marked in the leftmost polygon in Figure 4.9). We denote this angle by the same
symbol as the vertex. The complement of is the exterior angle at that vertex;
geometrically, it is the angle between one side and the extension of the adjacent side.
In any -gon, the sum of the angles equals ¾´ ¾µ right angles, or ¾´ ¾µ ¢ ¼ Æ ;
for example, the sum of the angles of a triangle is ½¼ Æ .
The area of a polygon whose vertices have coordinates ´Ü Ý µ, for ½
, is the absolute value of
Ö ½ ´Ü ¾ ½Ý ¾ Ü ¾ Ý ½ µ·¡¡¡· ½ ´Ü ¾ ½Ý Ü Ý ½ µ· ½ ´Ü ¾ Ý ½ Ü ½ Ý µ
½ (4.5.1)
´Ü Ý ·½ Ü ·½ Ý µ
¾
½
where in the summation we take Ü ·½ Ü ½ and Ý ·½ Ý ½ . In particular, for a
triangle we have
Ö ½ ¾ ´Ü ½Ý ¾ Ü ¾ Ý ½ · Ü ¾ Ý ¿ Ü ¿ Ý ¾ · Ü ¿ Ý ½ Ü ½ Ý ¿ µ ½ ¾
¬
¬
Ü ½ Ý ½ ½
Ü ¾ Ý ¾ ½
Ü ¿ Ý ¿ ½¬ (4.5.2)
In oblique coordinates with angle between the axes, the area is as given above,
multiplied by ×Ò .
If the vertices have polar coordinates ´Ö µ, for ½ , the area is the
absolute value of
Ö ½ ¾
½
Ö Ö ·½ ×Ò´ ·½ µ (4.5.3)
where we take Ö ·½ Ö ½ and ·½ ½ .
Formulae for speci c polygons in terms of side lengths, angles, etc., are given
on the following pages.
4.5.1 TRIANGLES
Because the angles of a triangle add up to ½¼ Æ , at least two of them must be acute
(less than ¼ Æ ). In an acute triangle all angles are acute. A right triangle has one
right angle, and an obtuse triangle has one obtuse angle.
The altitude corresponding to a side is the perpendicular dropped to the line
containing that side from the opposite vertex. The bisector of a vertex is the line
that divides the angle at that vertex into two equal parts. The median is the segment
joining a vertex to the midpoint of the opposite side. See Figure 4.10.
Every triangle also has an inscribed circle tangent to its sides and interior to
the triangle (in other words, any three non-concurrent lines determine a circle). The
center of this circle is the point of intersection of the bisectors. We denote the radius
of the inscribed circle by Ö.
Every triangle has a circumscribed circle going through its vertices; in other
words, any three non-collinear points determine a circle. The point of intersection of
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