Lune of Hippocrates - Generalizations

Generalizations

As Hippocrates showed using a similar proof to the one above, if two lunes are formed on the two sides of a right triangle, whose outer boundaries are semicircles and whose inner boundaries are formed by the circumcircle of the triangle, then the areas of these two lunes add to the area of the triangle. The quadrature of the lune of Hippocrates is the special case of this result for an isosceles right triangle. The lunes formed in this way from a right triangle are known as the lunes of Alhazen, named after the 10th and 11th century Arabic and Persian mathematician Alhazen.

In the mid-20th century two Russian mathematicians, Nikolai Chebotaryov and his student Anatoly Dorodnov, completely classified the lunes that are constructible by compass and straightedge and that have equal area to a given square. All such lunes can be specified by the two angles formed by the inner and outer arcs on their respective circles; in this notation, for instance, the lune of Hippocrates would have the inner and outer angles (90°,180°). Hippocrates found two other squarable concave lunes, with angles approximately (107.2°,160.9°) and (68.5°,205.6°). Two more squarable concave lunes, with angles approximately (46.9°,234.4°) and (100.8°,168.0°) were found in 1766 by Martin Johan Wallenius and again in 1840 by Thomas Clausen. As Chebotaryov and Dorodnov showed, these five pairs of angles give the only constructible squarable lunes; in particular, there are no constructible squarable convex lunes.

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