The problem can be reduced to that of finding the minimal segment B

The minimal secant (called sometimes

To solve the problem we consider an arbitrary secant BC, and think of the angles at B and C depending on one parameter. Denoting the angles by the same letters and differentiating the relation B+C+O = pi => B'+C' = 0 (angle O is constant). On the other side we have to minimize BC = BA+AC = AB

(AB

But D being the projection of D on BC we have the relation OD=DB*tanB=DC*tanC => DB*cotC-DC*cotB = 0. Thus, if BC is minimal A, D divide BC in the same ratio but in opposite directions, hence we must have AC = BD and D must coincide with D

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