Consider a circular arc, radius $r$, with endpoints $A$ and $D$. Arc $AD$ subtends an angle $\varphi$ < $\pi$. Thus, the arc length of $AD$ is $s_{AD} = r\varphi$. Place two points, $B$ and $C$ on $AD$ such that the arc measure $m(AB)$ is equal to the arc measure $m(CD)$. Let the angle subtended by $BC$ be $\theta$. Then, the arc length of $BC$ is $s_{BC} = r\theta$.
It is trivial to show that if the arc measure $m(BC)$ is $80$% that of $m(AD)$, then $5\theta = 4\varphi$. My question is as follows:
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Consider the linear segments $\overline{BC}$ and $\overline{AD}$. Knowing the above relationship between arc lengths such that $5\theta = 4\varphi$, is there a direct relationship between the linear measures, i.e. does $\dfrac{m(\overline{AD})}{m(\overline{BC})}$ have a definition?
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I have come to a few conclusions but I am not able to entirely eliminate the mention of the angles involved. I am seeking a direct relationship between linear measures.
Thank you.