How to prove that these ratios are the same?

Question

---

As we know, the measures of central angles of the arcs of a circle is proportional to the lengths of the arcs

How to prove that those ratios are the same?

Answer 1

In order to follow Euclid's proof you would need to go through a vast amount of development. Modern treatments will get you the answer much quicker.

• The length of a curve is defined as follows:
  • choose a finite number of points on the curve, in the order they appear on it. Add up the distances from each point to the next.
  • Take the set of all these sums for all possible finite sets of points on the curve.
  • The length of the curve is the supremum (least upper bound) of that set. Note that it may be infinite (such as for fractal or unbounded curves).
• An angle is a figure consisting two rays with a common end-point. The common endpoint is called the vertex, and the two rays are called the sides. This is effectively Euclid's definition in modern language (almost - Euclid's "lines" included curves). You will sometimes see generalizations of this definition allowing for negative angles or angles of greater than $180^\circ$ or even $360^\circ$, but I'll stick with the simple definition here. When one talks about the angle between two line segments with a common endpoint, the angle itself is the one formed by extending the segments to rays.
  • There is a unique plane containing an angle (unless the two rays lie on the same line, a special case I won't discuss here). If you extend a side to a full line, the line breaks the plane into two half-planes. The other side will in one of those half-planes. Take that half-plane for each of the two sides and intersect them. The region of intersection is called the interior of the angle.
  • Draw a circle of radius $1$ about the vertex. The two sides break the circle into two arcs, one of which lies in the interior of the angle (we say that this arc is subtended by the angle). The radian measure of the angle is the length of this subtended arc. The measure in degrees is just the radian measure multiplied by $\frac {180}\pi$.
  • The ratio of two angles is the ratio of their radian measures.
• Let $\theta$ be that radian measure. Draw another circle of radius $r$ about the vertex and consider the arc of this circle subtended by the angle. If we choose a sequence of points on this arc and draw the rays from the center to each of these points, they form a sequence of angles. Each ray also intersects the unit circle, the intersections of which form a sequence of points on that arc. Connect the adjacent points on each arc with chords. For each angle, you get a triangle for each circle. These triangles share the angle at the center, and the two sides of that angle are radii of the circle, so the ratios of corresponding sides are both $r:1$. So the two triangles are similar, and the remaining sides - the chords of the two circles - must also be in ratio $r:1$. If we sum the lengths of all the chords for each arc, those sums must also be in ratio $r:1$. So when calculating the lengths of the two arcs per the definition of length above, each sum for the arc of radius $r$ will be exactly $r$ times a sum for the arc of radius $1$. And each sum for the arc of radius $1$ will likewise be $\frac 1r$ times a sum for the arc of radius $r$. Ergo, the lengths of these two arcs - the suprema of the two sets of sums - must also be in ratio $r:1$. Since the length of the arc of radius $1$ is $\theta$, the radian measure of the angle, the length of arc of radius $r$ is $r\theta$.
• Finally in your case, you have two angles with measures $\theta$ and $\phi$ (which is $1$ in your picture, but I'll be a little more general). Because the arcs subtended by those two angles are on the same circle of radius $r = OA = OB = OC$, the lengths of the two arcs are $r\theta$ and $r\phi$. So the ratios in question are $$\frac\theta\phi\quad\text{and}\quad\frac{r\theta}{r\phi}$$ which are the same.