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
then since A and B are on the unit circle,
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
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.
next theorem (5.6)