The photo probably describes my concern best.I require to find a way to divide a circle into 3 parts of equal area with just 2 present that crossing each other on the outline of the circle.Also I should check, if whatever diameter is in between those lines, additionally splits circles with a different diameter into equal parts.And lastly, and probably the most challenging question: just how do I have to calculate the angle in between x lines that all intersect in one point, so the the one is break-up into x+1 parts with area = 1/(x+1) the the circle?I tried my best, but couldn"t even find a single answer or the best strategy to handle the question...

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edited Mar 18 in ~ 20:50
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Andrei
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request Mar 18 at 20:40
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JonasHausJonasHaus
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See more: Wilson In The Red Badge Of Courage What Happens To Wilson, The Red Badge Of Courage Chapters 13

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Given the edge $ heta$, split by the diameter comprise $B$, consider the complying with diagram:

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$overlineBO$ is the line v the center and $overlineBA$ is the chord cutting turn off the lune whose area we wish come compute.

The area the the one wedge subtended by $angle BOA$ is$$fracpi- heta2r^2 ag1$$The area of $ riangle BOA$ is$$frac12cdotoverbracersinleft(frac heta2 ight)^ extaltitudecdotoverbrace2rcosleft(frac heta2 ight)^ extbase=fracsin( heta)2r^2 ag2$$Therefore, the area that the lune is $(1)$ minus $(2)$:$$fracpi- heta-sin( heta)2r^2 ag3$$To acquire the area separated into thirds, we want$$fracpi- heta-sin( heta)2r^2=fracpi3r^2 ag4$$which method we desire to solve$$ heta+sin( heta)=fracpi3 ag5$$whose solution deserve to be completed numerically (e.g. Use $M=fracpi3$ and also $varepsilon=-1$ in this answer)$$ heta=0.5362669789888906 ag6$$Giving us

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Numerical Details