In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry (Euclidean, hyperbolic and elliptic), with those axioms of congruence that involve the concept of the angle dropped, and `triangle inequality', regarded as an axiom, added. If one assumes the continuity axiom in addition, then, in the case of the Euclidean plane, we come to the problem posed by Jean Gaston Darboux: "To determine all the calculus of variation problems in the plane whose solutions are all the plane straight lines." There are several interpretations of the original statement of David Hilbert. Nevertheless, a solution was sought, with the German mathematician Georg Hamel being the first to contribute to the solution of Hilbert's fourth problem. A recognized solution was given by Ukrainian mathematician Aleksei Pogorelov in 1973.<ref name="Pogorelov1973">А. В. Погорелов, Полное решение IV проблемы Гильберта, ДАН СССР № 208, т.1 (1973), 46–49. English translation: {{cite journal | last1=Pogorelov | first1=A. V. | title=A complete solution of "Hilbert's fourth problem| journal=Doklady Akademii Nauk SSSR | volume=208 | issue=1 | date=1973 | pages=48—52}}</ref> In 1976, Armenian mathematician Rouben V. Ambartzumian proposed another proof of Hilbert's fourth problem. Original statement Hilbert discusses the existence of non-Euclidean geometry and non-Archimedean geometry ...a geometry in which all the axioms of ordinary euclidean geometry hold, and in particular all the congruence axioms except the one of the congruence of triangles (or all except the theorem of the equality of the base angles in the isosceles triangle), and in which, besides, the proposition that in every triangle the sum of two sides is greater than the third is assumed as a particular axiom. Due to the idea that a 'straight line' is defined as the shortest path between two points, he mentions how congruence of triangles is necessary for Euclid's proof that a straight line in the plane is the shortest distance between two points. He summarizes as follows: The theorem of the straight line as the shortest distance between two points and the essentially equivalent theorem of Euclid about the sides of a triangle, play an important part not only in number theory but also in the theory of surfaces and in the calculus of variations. For this reason, and because