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The altitudes of a triangle

Mark Mandelkern

Abstract

A long-standing, unanswered question regarding Euclid's Elements concerns the absence of a theorem for the concurrence of the altitudes of a triangle, and the possible reasons for this omission. In the centuries following Euclid, a remarkable number of proofs have been put forward; this suggests a search for the most elementary and direct proof. This paper provides a simple, direct, elementary proof of the theorem; it is based solely on the Elements.

The altitudes of a triangle

Abstract

A long-standing, unanswered question regarding Euclid's Elements concerns the absence of a theorem for the concurrence of the altitudes of a triangle, and the possible reasons for this omission. In the centuries following Euclid, a remarkable number of proofs have been put forward; this suggests a search for the most elementary and direct proof. This paper provides a simple, direct, elementary proof of the theorem; it is based solely on the Elements.
Paper Structure (1 theorem, 1 figure)

This paper contains 1 theorem, 1 figure.

Key Result

Theorem 1

The altitudes of a triangle are concurrent.

Figures (1)

  • Figure :

Theorems & Definitions (2)

  • Theorem
  • proof