Triangle their internal angles, measured here in degrees. A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a "2-simplex" (see also Polytope). Elementary facts about triangles were presented by Euclid in books 1–4
Napoleon points three triangles "XBC", "YCA", "ZAB" erected over the sides of the triangle "ABC" need not be isosceles for the three lines "AX", "BY", "CZ" to be concurrent. The concurrence of the lines "AX", "BY", and "CZ" holds even in much relaxed conditions. The following result states one of the most general conditions for the lines "AX", "BY", "CZ" to be concurrent. The point of concurrency is known as the Jacobi point. Coxeter and Greitzer state the Napoleon Theorem thus: "If equilateral triangles are erected externally on the sides of any triangle, their centers form an equilateral triangle". They observe that