The circle divides the plane into two regions: the *inside*,
which is the set of points at distance less than from the center ,
and the *outside*, which consists of all points having distance from
greater than . Note that every line segment from to a point on
has the same length .

A line segment from to a point on is also called a
*radius*; this should cause no confusion.

A line segment connecting two points of is called a *chord*,
if the chord passes through the center, then it is called a
*diameter*.

As above, we also use the word diameter to denote the length of a diameter of , that is, the number that is twice the radius.

We first take up the case that is a diameter. In this case, we would have at least two of the three points on the same side of on ; hence we can suppose that and both lie on the same side of . However, by the ruler axiom (Axiom 6), we must have since . This contradicts our assumption that and both lie on .

We next take up the case that is not a diameter. We can assume that lies between and on . Draw the line segments, , , . Then , and are triangles. In fact, since , they are isosceles triangles. Let be the measure of the base angles of triangle . Then it is also the measure of the base angle of , and so it is also the measure of the base angle of . Since the two base angles at add up to , we obtain that each of the three triangles have two right angles, which is impossible.

First, we assume the well-known property that the *circumference*
(that is, the arc length) of a circle of radius is .

Now let and be two points on a circle of radius 1 and center .
The radii and make two angles (the
``inner'' and the ``outer'' angles); call them and .
An angle such as whose vertex is at the center of the circle is
called a *central angle*. Notice that
, no
matter where and lie on .
If and are the endpoints of a diameter, they divide the circle into
two arcs, each of length ; note also that the measure of the angles
and are also . In other cases, the length of the arc
subtended by the angle will be whatever fraction of that
is of the entire circle. For example, if is a right
angle, it will take up of the circle, and the corresponding arc length
will be . We define the measure of the angle to be the corresponding
arc length when that angle is the central angle of a circle of radius 1.

Note that another way to describe a circle circumscribed about a triangle is to say that it is the smallest circle for which every point inside the triangle is also inside the circle. In this view, the problem of circumscribing a circle becomes a minimization problem. A given triangle lies inside many circles, but the circumscribed circle is, in some sense, the smallest circle which lies outside the given triangle.

It is not immediately obvious that one can always solve this minimization problem, nor that the solution is unique.

Let and be the midpoints of sides and respectively. Draw the perpendicular bisectors of and , and let be the point where these two lines meet (note that need not be inside the triangle). Draw the lines , and .

If , then we repeat the above argument to show that , from which, as above, it follows that . Again, this shows that there is a circumscribed circle.

We start the search for the inscribed circle with the question of what it means for the circle to have two tangents which are not parallel.

By *SSA*,
. Hence
.

From the above, we see that if there is an inscribed circle for , then its center lies at the point of intersection of the three angle bisectors, and its radius is the distance from this point to the three sides. Hence we have proven the following.

Observe that, by *AAS*,
. Similarly,
and
.

We have shown that the perpendiculars from to the three sides all have equal length; call this length . then the circle centered at of radius is tangent to the three sides of exactly at the points , and .

This theorem gives another proof of the result of exercise 2.16.