The picture probably defines my inquiry best.I need to find a means to division a circle into 3 components of equal area with only 2 present that crossing each other on the summary of the circle.Also I should check, if whatever diameter is in between those lines, additionally splits circles v a different diameter right into equal parts.And lastly, and also probably the most daunting question: just how do I have to calculate the angle between x lines that all crossing in one point, so that the one is separation into x+1 components with area = 1/(x+1) that the circle?I tried mine best, however couldn"t even uncover a solitary answer or the appropriate strategy to tackle the question...


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edited Mar 18 in ~ 20:50

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


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

The area that the one wedge subtended by $angle BOA$ is$$fracpi- heta2r^2 ag1$$The area that $ 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 split into thirds, us 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 achieved numerically (e.g. Use $M=fracpi3$ and $varepsilon=-1$ in this answer)$$ heta=0.5362669789888906 ag6$$Giving us


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