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Bisecting horn angles

A horn angle between a circle and its tangent is considered in Euclid's Elements, and Euclid remarks that it is smaller than any acute rectilinear angle. Already in antiquity, Proclus wondered whether it is possible to bisect horn angles. We will give a construction of a bisector which was within the means of ancient geometers since the time of Archimedes and Apollonius. We will also compare it to the conformal bisection method introduced in modern times.