Prove truth of circle division theory by 5 equal parts

mister_giga

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Hello. I have forgotten much of my math I guess. There is one theory that gives us the length of the five equal parts of the circle. I have to prove that this theory is true (it is true for sure but I still need to prove). Can you help me? circle5partProblem.jpg

I try to prove by going from both parts. I mean I count OK from the EB radius perspective and I equalize it with the 2*pi*R/10 but I dont get 1=1 at the end.

This is how I try to solve, I get x=0.

20171119_213857.jpg

Please help if you understand what is my problem
 
Your claim that OK is 1/10 of the circumference of the circle (which I would call C rather than S) is false. Perhaps it is approximately so, but it can't be exactly true, because we know that the circumference can't be constructed with compass and straightedge. Can you tell me the source of your claim? Who told you that this is true, or told you to prove it?

The construction looks very similar to the construction of a regular pentagon; in fact, your segment BK is equal to the side of the pentagon. But looking at the actual values of OK and C/2, they are

OK = R(sqrt(5) - 1)/2 = R*0.618..., as you found

C/10 = 2 pi R/10 = R*pi/5 = R*0.6283...

So, yes, it appears to be an approximation.
 
I cannot read the second picture, too small.

You have a circle with the origin at O and perpendicular diameters AOB and COD. The midpoint of AO is at E. You construct the right triangle EOD. The base has length r/2 and the height is r, giving a hypotenuse with length of

\(\displaystyle \sqrt{r^2 + \left ( \dfrac{r}{2} \right )^2} = \sqrt{\dfrac{5r^2}{4}} = 0.5r\sqrt{5} = p.\)

You now construct a circle with an origin at E and a radius of p. That circle intersects line segment OB at F. The length of OF is

\(\displaystyle p - 0.5r = q\)

I got that from the first picture. So what is the problem?
 
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