The Art and Craft of Problem Solving

8.2 SURVIVAL GEOMETRY I 267

It is not immediately obvious that an arbitrary triangle even has an inscribed or

circumscribed circle. Nor is it obvious that if such circles exist, that they be unique.

Let's explore these questions, starting with the circumscribed circle.

8.2. 19 Suppose a triangle possesses a circumscribed circle. Then the center of this

circle is the intersection point of the perpendicular bisectors of the sides of the triangle.

(Consequently, if there is a circumscribed circle, then it is unique.)

8.2.20 Given any triangle, consider the intersection of any two perpendicular bisectors

of its sides. This intersection point will be equidistant from the three vertices; hence

will be the center of the circumscribed circle.

8.2.2 1 Thus the circumcenter is the intersection point of all three perpendicular bisec­

tors. (So we get, "for free," the interesting fact that the three perpendicular bisectors

intersect at a single point.)

Likewise, there is a (unique) inscribed circle whose center is the intersection point

of three special lines. Again, our strategy is to first assume that the inscribed circle

exists, and explore its properties. The tricky part, as above, is uniqueness.

8.2. 22 Every triangle has a unique inscribed circle. The center is the intersection point

of the three angle bisectors of the triangle. (Hint: Suppose the inscribed circle exists.

Show that its center is the intersection of two angle bisectors, using Fact 8.2.1 4. Then

show that the converse is true; i.e., the intersection of any two angle bisectors is the

center of an inscribed circle. Finally, show that there cannot be more than one such

circle, and hence all three bisectors meet in a single point.)

The existence of the circumcircle and incircle led to the wonderful facts that in

any triangle, the three angle bisectors are concurrent (meet in a single point), and

the three perpendicular bisectors of the sides are concurrent. There are many other

"natural" lines in a triangle. Which of them will be concurrent?

We will explore this is greater detail in Section 8.4, but here is a nice example that

uses an ingenious auxiliary construction, and not much else.

Example 8.2.23 Show that, for any triangle, the three altitudes are concurrent. This

point is called the orthocenter of the triangle.

Solution: An altitude is a line (not a line segment) that passes through a vertex
and is perpendicular to the opposite side (extended, if necessary). Informally, the word
refers to a line or a segment, or a segment length , depending on context. For example,

in the figure below, the altitude from B to side AC is the actually the line BD, but

it is not uncommon for the segment BD or its length (also written BD) to be called

the altitude. The point D at which the altitude intersects the side is called the foot

of the altitude. In standard usage, the foot is not explicitly mentioned; for example,

"altitude BD" instead of the more precise "drop a perpendicular from vertex B to side

AC, intersecting it at D."

Notice that altitudes may not always lie inside the triangle. The altitude through C

meets the extension of side AB at E, outside the triangle. Notice also that altitudes CE