Concurrency of the Altitudes of a Triangle

Of all the traditional (or Greek) centers of a triangle, the orthocenter (i.e., the point of concurrence of the altitudes) is probably the one that attracted the most of attention. This may be due to the fact that it is the only one that has no exact analogue for arbitrary higher dimensional simplices, for spherical and hyperbolic triangles, or for triangles in normed planes. But it possibly has to do also with the non-existence of any explicit treatment of this center in the Greek works that have come down to us. In this paper we present different proofs of the fact that the altitudes of a triangle are concurrent. These include the first extant proof, in the works of al-Kūh̄ı, Newton’s proof, Gauss’s proof, and other interesting proofs.