Let be a regular -gon with points . Fix one point and define as the distance from to . EvaluateProblem(1):

What about ?Problem(2):

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- December 3rd 2009, 05:14 PMDrexel28Geometry(2)
Let be a regular -gon with points . Fix one point and define as the distance from to . Evaluate__Problem(1):__

What about ?__Problem(2):__ - December 4th 2009, 05:09 AMUnbeatable0
__Spoiler__: - December 4th 2009, 11:00 PMDrexel28
- December 4th 2009, 11:09 PMsimplependulum
Suppose it has a unit radius ( circumscribed circle)

(1) :

(2) : - December 5th 2009, 05:27 AMDrexel28
- December 5th 2009, 06:04 AMsimplependulum
- December 5th 2009, 07:35 AMUnbeatable0
I have to say, simplependulum - your solution is very inspiring! (Clapping)

So to sum it up, in the general case we have:

Where is the radius of the circumscribed circle.

Thank you Drexel28 for this interesting problem with surprisingly simple solution formulas. (Nod)

Edit:

After noticing that the last expression for the product of the distances does not include trigonometry, an interesting question came into my head: can you find this formula without the use of trigonometry? - December 7th 2009, 10:00 PMDrexel28
- December 27th 2009, 01:28 AMUnbeatable0
I'm interested to see the proof you referred to.

Will you please post the proof? Or at least the idea of the proof (using roots of unity does not lead me too far).

Thanks in advance (Happy)

Too bad, I couldn't send you a private message because my post count is not above 15.