Pascal's triangle Construct a "d"-dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to P(1) = 1. Place these dots in a manner analogous to the placement of numbers in Pascal's triangle. To find P("x"), have a total of "x" dots composing the target shape. P("x") then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore P("x") = "1" and P("x") = "x", which is the sequence of natural numbers. The number of dots in each layer corresponds
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