to reach the same conclusion that (or in this case ) must be the Fermat point of . Lagrange multipliers Another approach to finding the point within a triangle, from which the sum of the distances to the vertices of the triangle is minimal, is to use one of the mathematical optimization methods; specifically, the method of Lagrange multipliers and the law of cosines. We draw lines from the point within the triangle to its vertices and call them . Also, let the lengths of these lines be respectively. Let the angle between and be , and be . Then the angle between and is . Using the method of Lagrange multipliers we have to find the minimum of the Lagrangian , which is expressed as: where are the lengths of the sides of the triangle. Equating each of the five partial derivatives to zero and eliminating eventually gives and so . However the elimination is a long and tedious business, and the end result covers only Case 2. Properties When the largest angle of the triangle is not larger than 120°, X(13) is the Fermat point. The angles subtended by the sides of the triangle at X(13) are all equal to 120° (Case 2), or 60°, 60°, 120° (Case 1). The circumcircles of the three constructed equilateral triangles are concurrent at X(13). Trilinear coordinates for the first isogonic center, X(13): Trilinear coordinates for the second isogonic center, X(14): Trilinear coordinates for the Fermat point: where respectively denote the Boolean variables . The isogonal conjugate of X(13) is the first isodynamic point, X(15): The isogonal conjugate of X(14) is the second isodynamic point, X(16): The following triangles are equilateral: antipedal triangle of X(13) Antipedal triangle of X(14) Pedal triangle of X(15) Pedal triangle of X(16) Circumcevian triangle of X(15) Circumcevian triangle of X(16) The lines X(13)X(15) and X(14)X(16) are parallel to the Euler line. The three lines meet at the Euler infinity point, X(30). The points X(13), X(14), the circumcenter, and the nine-point center lie on a Lester circle. The line X(13)X(14) meets the Euler line at midpoint of X(2) and X(4). The Fermat point lies in the open orthocentroidal disk punctured at its own center, and could be any point therein. Aliases The isogonic centers X(13) and X(14) are also known as the first Fermat point and the second Fermat point respectively. Alternatives are the positive Fermat point and