Is it possible using only a straight edge and a compass to construct a Venn Diagram or a pair of overlapping circles in which the area of the lune of each circle is equal to the other and to the area of the overlapping region?(adsbygoogle = window.adsbygoogle || []).push({});

(I've worked out most of the mathematics from calculating the area of the sector (cad) and subtracting the area of the triangle (cad) to achieve the segment of the circle created by the chord (cd) which accounts for half the total overlap. Then to achieve a useful equation, I calculated that, obviously, the area of the circle divided in half must equal twice the area of the single segment created by the chord.)

After simplifying, it would seem that to construct the required angle θ the equation

θ-sin(θ)=π/2

would have to be solved (in radians), which would seem to denote that it is not a constructible angle.

But questions of constructibility usually require someone much more clever than I, so am I missing a simplifying way to achieve this? and is it fair to say that the only way to solve the above equation is through the use of numerical methods?

*if it would be useful to review the full calculation, I can post that as well- it's just a bit tedious

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# Constructibility of a Perfect Venn Diagram

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