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I honestly do not know why I'm so lost here, but I am. Really, I'm proving the construction of an equilateral triangle inscribed in a circle, but the inscribed regular hexagon is crucial to that proof (at least the way I'm proving). There are proofs online, but they are not satisfying to me. Here is my question. So, we construct a circle, then label any point P on the circle, then we swing 6 arcs across the circumference of the circle starting at point P and end on our original point while keeping the compass set to the length of the radius of the circle. Does this process not imply that the circumference of a circle is 6 times the radius? We know that this is not true... what am I missing here?

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    $\begingroup$ Does this process not imply that the circumference of a circle is 6 times the radius No, why would it? What it does imply is that the perimeter of the regular hexagon is 6 times the radius. $\endgroup$ – dxiv Jan 23 '18 at 4:19
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    $\begingroup$ Was still initially confused with this comment, but now I see. The chord that connects each of those points is equal to the radius instead of the curved distance between each point... sheesh, thank you. $\endgroup$ – MathGuy Jan 23 '18 at 4:27
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No, it does not mean anything about the circumference of the circle. You are not measuring the circumference with your compass.

You just measure the cord not the the length of the arc.

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  • $\begingroup$ Yes, I just now noticed that (see comment above). Duh... thank you. $\endgroup$ – MathGuy Jan 23 '18 at 4:28
  • $\begingroup$ @TyeCampbell Thanks for attention. $\endgroup$ – Mohammad Riazi-Kermani Jan 23 '18 at 4:50

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