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: Reading Archimedes, I noticed a strange omission wonder if this group would like to have a look at this question. : In On the Sphere and the Cylinder I (page 11 of the Dover reprint of the Heath translation, now happily back in print), Archimedes has occasion to look at three triangles making a trihedral angle. These are formed as follows. In an isosceles cone (aka as a right circular cone) with apex O, a chord AB is drawn in the base. Then from the midpoint (say C) of one of the arcs cut by the chord AB, the lines AC and BC are drawn, as are the lines OA, OB, and OC. Now Archimedes claims, without proof, as though it were obvious, that: triangle OAB < triangle OBC + triangle OAC . I’ll call this “claim one”. : Now I definitely have a proof, due to Klamkin, (see attachment) of “claim one” (in a more general form) but I’m writing to ask if anyone can find a geometric proof of this fact and/or if you have seen this fact proved in either Euclid or Archimedes or anywhere else. The proof I have uses some trig identities. Of course these are ultimately geometric but I’m thinking that if Archimedes had my proof in mind (very, very doubtful), he would have shown it. :

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