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

A = (x1, y1)

and

B = (x2, y2)

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  

rA = (rx1, ry1)

and

rB = (rx2, ry2)

the distance from   rA   to the origin is

= r

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

= r|AB|

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.

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