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Hey I have the following question below:

enter image description here

I want to ask is it necessary to include the arc symbol (the frown) above BC and CD. If the question was BC = 2CD, BA = BD is this still mathematically correct? Is the symbol just syntactic sugar? I'm asking this because I am building an iOS app and the latex library doesn't support this symbol so want to avoid using it if possible. Any pointers on this would be great!! Thanks!

Kex
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  • In this case, the constraint is on the arcs of the circle, the arc between $B$ and $C$ is twice as long as the arc between $C$ and $D$. If you constrain the straight line segments, you get a different problem. You can use something like $\operatorname{arc}(B,C)$, that's clearer than those arcane symbols anyway. – Daniel Fischer Jun 11 '17 at 12:53
  • arc(B, C) = 2arc(C, D) would that be clear? – Kex Jun 11 '17 at 13:03
  • I think it would. – Daniel Fischer Jun 11 '17 at 13:05

1 Answers1

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The answer is NO.

In the figure, X is the midpoint of the arc BC.

enter image description here

Then, $\dfrac {arc (BC)}{arc (CX)} = \frac {r(4\theta)}{r(2\theta)} = \dfrac 21$.

But, $\dfrac {Chord (BC)}{Chord (CX)} = \dfrac {(2 \times r \sin 2 \theta)}{ 2 \times r \sin \theta} = \dfrac {\sin 2 \theta}{\sin \theta} \ne \dfrac 21$, in general.

Simply put:- If the arc is doubled, the corresponding chord is NOT doubled.

Another simple observation is:-

"CX, being the hypotenuse of a right angle triangle, is longer than the leg whose length is $\dfrac 12 BC$".

Mick
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