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
Equilateral triangle to the points where the angle bisectors of ∠"APB", ∠"BPC", and ∠"CPA" cross the sides ("A", "B", and "C" being the vertices). For any point P in the plane, with distances "p", "q", and "t" from the vertices A, B, and C respectively, For any point P on the inscribed circle of an equilateral triangle, with distances "p", "q", and "t" from the vertices, and For any point P on the minor arc BC of the circumcircle, with distances "p", "q", and "t" from A, B, and C respectively, and moreover, if point D on side BC divides PA into