Algebraic Adjustment of Measured Triangle Elements

It is difficult to align the measured elements of the triangle. A method of alignment using algebraic circles is proposed. If six elements are measured in a triangle: angles A, B, C, lengths of opposite sides a, b, c with mean square errors mA, mB, mC, ma, mb, mc, the values of the equalized elements must satisfy the requirements of the equations where D is the metric parameter of the triangle (the metric value through which the mathematical relationship between the sides and angles of the triangle is established), that is, the value of the diameter of the circle. If , which are found by formulas (1) and condition (2) is fulfilled, then this confirms the validity of the theorem of sines. In fact, it is a symmetrical set, because the number and composition of the upper indices corresponds to the number and composition of the lower indices, and this is a reflection of dimensions, since the triangle ABC has pairs of opposite elements: а- А; b- B; c- C. Let's construct the graph G Do of the diameter of the circle circumscribed around the aligned triangle, for which we will first construct the graphs G Di of the diameters of the circles circumscribed around the T-triangles, which are shown in fig. 1, and then combine them. Three vertices a, b, c form a "plane" - this is a metric field. The reflection of the metric field is the field of angular measurements - a "plane" defined by three points A, B, C. Note that for a measured triangle, the reflection of the metric field in the angular one is not unambiguous, so the first plane in fig. 1 is "inclined" to the second. After determining the most likely metric parameter Do of the triangle, it is possible to bring these planes into full correspondence. That is, Do is a "lens" 4,8 with the help of which we obtain an unambiguous mapping of the metric field in the angular field and vice versa.