Forum Geometricorum Volume 17 (2017) 185–195. FORUM GEOM ISSN 1534-1178 Triangle Constructions Based on Angular Coordinates Thomas D. Maienschein and Michael Q. Rieck Abstract. Two very different, yet related, triangle constructions are examined, based on a given reference triangle and on a triple of signed angles. These pro- duce triangles that are in perspective with the reference triangle and with each other, using the same center of perspective. The first construction is rather well- known, and produces a Kiepert-Morley-Hofstadter-Kimberling triangle. A new circumconic is associated with this construction. The second construction gen- eralizes work of D. M. Bailey and J. Van Yzeren. A number of known central triangles are obtainable using one or both of these constructions. 1. Introduction This article is concerned with two very different triangle constructions based on a given reference triangle. Each of these is also based an a triple of signed angles (ψ 1 ,ψ 2 ,ψ 3 ). These two constructions produce triangles that are in per- spective with the reference triangle and with each other, using the same point of perspective. If it happens that ψ 1 + ψ 2 + ψ 3 ≡ 0 (mod π), then the point of per- spective will just be the point whose angular coordinates are (ψ 1 ,ψ 2 ,ψ 3 ). The first construction is rather well-known, and produces the Kiepert-Morley-Hofstadter- Kimberling (KMHK) triangle, with ψ 1 , ψ 2 and ψ 3 serving as the swing angles. The second construction generalizes work of D. M. Bailey [1] and J. Van Yzeren [7]. It focuses attention on a certain triple of circles, where each circle passes through two of the reference triangle vertices. Section 2 carefully introduces the notions of “directed angles” and “angular co- ordinates,” in the sense in which we will be using these phrases. Section 3 details the construction of a Kiepert-Morley-Hofstadter-Kimberling triangle. Most of this material is admittedly already presented adequately in Chapter 6 of [4]. However, there is a result at the end of Section 3 here that appears to be new. Section 4 details our extension of [1] and [7], and this results in the construction of another triangle, as mentioned earlier. In Section 5, straightforward methods are presented for testing the trilinear coor- dinates of a given triangle to determine whether or not it can be obtained by means of one of the two constructions. In Section 6, the results of thus testing the exam- ples of central triangles in [4] are presented. Many of these central triangles passed Publication Date: June 6, 2017. Communicating Editor: Paul Yiu.