Development and prospects of Stewart’s theorem research

This paper is devoted to the study of Stewart's theorem, its consequences, development and prospects of research of the theorem under consideration. The paper focuses on the Diophantine equations and their relation to the Stewart’s theorem. The problem of determination of integer solutions of the Diophantine equations was considered, and some modern researches of the Stewart’s theorem are presented from the point of view of finding integer solutions of it, which are related to the first and second order Diophantine equations. The wellknown integer solutions of the Stewart’s theorem, and the definitions, which have been formulated in the form of a table, are presented in this paper. Some practical applications of the Stewart’s theorem focused on computing the length of a segment, that connects the vertex of a triangle with its inner point, are relevant in the area of logistics, management and designing.


INTRODUCTION
Stewart's theorem is one of the classical problems of triangle geometry and is partially represented in the elementary geometry educational literature [7,11]. The most well-known problem-consequences of the Stewart's theorem are formulas for calculating the lengths of the medians and bisectors of a triangle on its sides [2]. The consequence of the Stewart's theorem is the Ptolemy theorem, and the Apollonian theorem that is the partial case of Ptolemy theorem [8]. We have systematized problems-consequences and proofs of the theorem and set task to move away from the classical method of the topic presentation and to characterize modern directions of researches of the theorem and to find its relations with other sections of mathematics. The question of finding an integer solution of the Stewart equality as a search for the solution of the Diophantine equation remains interesting.

MATERIAL AND METHODS
The purpose of the paper is to formulate the Stewart's theorem and its development, to present modern studies of the theorem under consideration. We refer the integer solutions of the Stewart's theorem and show the relation to the Diophantine equations. The research methods used in this paper are universally recognized methods of scientific knowledge [10]: -Theoretical: study and analysis of relevant scientific literature, textbooks and materials of electronic publications; -Inductive: collection, systematization and classification of existing proofs of the Stewart's theorem, its consequences and current studies of the theorem under consideration; -Practical: application of theoretical knowledge and practical skills to create problems and their solutions.

RESULTS AND DISCUSSION
Scottish mathematician Stewart Matthew (1717 − 1785) published a scientific paper in 1746 ( Fig. 1) in which he presented his theorem with substantiation [15]. This theorem is called Stewart's theorem, and is expressed by (1) and displayed in Fig. 2.

Figure 1
Stewart's theorem [20]. Let ΔАВС be an arbitrary triangle (Fig. 2). For any point D on the side of the BA the following formula holds: (1)   There are various proofs of Stewart's theorem: by Pythagorean Theorem, coordinate method,  cosine theorem, vector method [1, 2,8]. The work [2] presents the consequences of the Stewart's theorem: finding the length of a segment with ends on the sides of a triangle with possible boundary cases, and finding the distance from the vertex to the inner point of the triangle. As a development of the Stewart's theorem Willie Yong and Jim Bound in their paper Using Stewart's Theorem [20] presents the solution of the following problem: ( Consider that TA BS = , by the Stewart's theorem we have: ( Substituting (3) into (2), we get: In view of (3), (4) we subtract and obtain: One of the problems that were formulated by Willie Yong and Jim Bound is presented below Problem 1. Prove that in the right triangle the sum of the squares of distances from the vertex of the right angle to the three points of hypotenuse is equal to 9 5 of the length of the hypotenuse squared, i.e., АB CT CS ⋅ = + 9 5 2 2 . The solution was presented in [17].

Solution.
In view of the Stewart's theorem the following equalities hold (Fig. 4): We divide the right hand sides of the above equations with a and get . What was to be shown.
Bretschneider's formula describes the relation among the elements of the quadrilateral ABCD and is related to the Stewart's theorem. Let us denote (Fig. 5) Theorem (Bretschneider's formula). The following formula holds Proof. We construct two triangles ABF and ADE, which are similar to the triangles CAD and CAB respectively. The similarty of the triangles yields to In case 0 , we have that one of the factors is zero.
In other words 0 , then we choose the value so that 2 с could be completely divided by а.
We find 2 b such that a c a b 2 2 ⋅ is a square of some integer and is equal to 2 p , this could be rewritten as 0 Results of the applied selection method are presented in Table 1.  We find the values of у, such that is an integer or ay b − 2 can be factored by а.
As a result we get: Thus, after calculations we have computed four different integer solutions of the Stewart's equation in a form that corresponds to the equilateral triangles in notations of Fig. 6 (Fig. 7).

CONCLUSIONS
In this paper the Stewart's theorem was considered as one of the classical problems of geometry of a triangle and its application to different problems. We studied the problem of computing integer solutions of Diophantine equations and presented some results of studies of the Stewart's theorem in view of finding its integer solutions, that is related to first-and second-order Diophantine equations. The method for computing integer solutions of Stewart's equations was presented and four different integer solutions for equilateral triangles were computed, using the described method.
From the obtained results we can conclude that application of the Stewart's theorem is an effective tool for solving numbers of geometrical problems. The generalizations of the Stewart's theorem and its applications to computing the length of a segment, that connects the vertex of a triangle with its inner point, can be applied in logistics, management and designing. For example, to choose the right coordinates for set of three stores, which are the vertices of the triangle or to find correct place for Wi-Fi router for a few settlements (for three, for example).