Central Points and Central Lines in the Plane of a Triangle

The uses of homogeneous barycentric coordinates in plane Euclidean geometry

“…Thus, X 37ˆI ¢ P, where P is the point with homogeneous barycentric coordinates b ‡ c : c ‡ a : a ‡ b. This happens to be the point X 10 of [2], the Spieker centre of the triangle, the incentre of the triangle formed by the midpoint of the sides of the given triangle. Without relying on this piece of knowledge, a direct, simple calculation with the barycentric coordinates leads to an easy construction of this point.…”

“…It can therefore be located as the intersection of the two lines IK and GG e . The fact that X 9 lies on both of these lines is stated in [2], along with other lines containing the same point.…”

“…Consider, for example, the`simplest unnamed centre' X 37 in [2], with trilinear coordinates b ‡ c : c ‡ a : a ‡ b. In homogeneous barycentric coordinates, this is a…b ‡ c † : b…c ‡ a † : c…a ‡ b †.…”

“…Note that an easy application of Proposition 1 shows that the line joining P, G and I ¡1 does not contain I unless the triangle is isosceles. In fact, many of the collinearity relations in [2,3] can be explained by manipulating homogeneous barycentric coordinates. We present a few more examples.…”

“…With respect to a ® xed triangle ABC (of side lengths a, b, c, and opposite angles ¬, , ® ), the trilinear coordinates of a point is a triple of numbers proportional to the signed distances of the point to the sides of the triangle. The late Jesuit mathematician Maurice Wong has given [5] a synthetic construction of the point with trilinear coordinates cot ¬ : cot : cot® , and more generally, in [4] points with trilinear coordinates a 2n x : b 2n y : c 2n z from one with trilinear coordinates x : y : z with respect to a triangle with sides a, b, c. On a much grandiose scale, Kimberling [2,3] has given extensive lists of centres associated with a triangle, in terms of trilinear coordinates, along with some collinearity relations.…”