Concurrency of the altitudes of a triangle

Proving Concurrency by Loci

“…The above proof does not appear among the 12 proofs for the concurrency of the altitudes listed in [1].…”

“…It is easy to see that Proof #7 in [1] actually infers the concurrence of the altitudes from Carnot's concurrency theorem. As a matter of fact, the altitudes are loci of some other, trigonometric, kind.…”

“…It actually shows that the altitudes of a triangle are perpendicular side bisectors of another triangle, and so are concurrent (being the loci of some points). Two other proofs of this theorem (Proofs #4 and #5 in [1]) show that the altitudes of a triangle are internal angle bisectors of some other triangles. Here, we prove the concurrency of the altitudes by directly showing them to be some loci.…”

“…The concurrency of the altitudes and the medians do not appear in the Elements of Euclid, though they are classical theorems by now. Some believe that Archimedes knew the concurrency of the medians (see [3, p. 84]), and two proofs for the concurrency of the altitudes are attributed to Newton (1642-1726) and Gauss (1777-1855); see Proofs #2 and #1 in [1]. In this paper, we prove these theorems, and also the concurrency theorems of G. Ceva (1647-1734) and L. Carnot (1753-1823), by using loci.…”

“…In its proof, we have considered the case where the altitude lies inside the triangle (cf. [1,Lemma]); other cases can be dealt with similarly.…”