**Theorem
5.5**: If a
circle is drawn at the
vertex of an
angle, then the
arc length between the
points on the
circle where the arms of the angle meet the
circle is proportional
to the radius of the
circle.

**Proof**: Since arc
lengths are preserved by invertible
isometries, by
Theorem 5.3, we can
translate the
vertex of the
angle to the origin.

Let *A *and *B *be the points where the
angle meets the
unit circle. If

and

then since *A *and *B *are on the unit circle,

and

Since the ray of the
angle which includes the
point *A* goes through the
origin, we can conclude that the point *rA *is also on the ray . Similarly, *rB *is
on the ray of the
angle which contains the
point *B*. So since

and

the distance from *rA *to the origin is

This says that *rA *is on the circle centered at the
origin of radius* r*. This sets up a one to one correspondence between the partitions of the
arc on the
unit circle between *A *and *B *and the partitions of the arc on the circle centered
at the origin of
radius *r *between *rA *and *rB*. Moreover, the distance between *rA *and *rB *is

We conclude that for every
partition of the
arc on the
unit circle between *A *and *B*, there exists a partition of
the arc on the circle
centered at the origin of
radius *r *and the inside
approximation for that
partition will be *r *times the inside
approximation for the arc
length along the unit
circle. We conclude that, in the limit, the
arc length from *rA *to *rB *will
be *r *times the arc length
from *A *to *B*, or the ratio of the arc length from *rA *to *rB *to
arc length from *A *to *B*, will be *r *for
all positive values of *r*.