Triangles I: Shapes

Abstract

Summary. This paper is the first in a series of three examining Euclidean triangle geometry via complex cross ratios. In this paper we show that every triangle can be characterized up to similarity by a single complex number, called its shape. We then use shapes and two basic theorems about shapes to prove theorems about similar triangles. The remaining papers in this series will examine complex triangle coordinates and complex triangle functions. IntroductionThe standard identification of the classical Euclidean plane with the Argand, or Gaussian plane provides a useful setting for the development of many aspects of Euclidean geometry in terms of complex numbers and their properties. ( [19].) An important feature of this identification is that many of the metric and incidence properties of the Euclidean plane can be formulated in terms of complex cross ratios (as developed, for example, in [22]). This paper and its sequels [9], [10] develop the geometry of triangles and their special points through a complex analytical formalism based on cross ratios.In this first paper, we are interested in analytic methods for proving similarity theorems. After some notation and mathematical preliminaries in §2, we observe in §3 that, using cross ratios, any Euclidean triangle may be classified up to similarity by a single complex number. (As it turns out, this fact was also noted and used as early as 1900 [19], and reappears on occasion in more recent work, for example [13].) We call this number the shape of the triangle, and, once we develop some of its basic properties, move on in §4 to discuss the shapes of Miquel triangles. The AMS (1991) subject classification: 51M05, 51N20. January 22, 1993 and, in final form, February 8, 1995. 30 Vol. 52, 1996 Triangles I: Shapes 31 latter part of the paper ( § §5, 6) then examines two general shape theorems and illustrates each with several examples of theorems it can be used to prove. The latter shape theorem ( §6) has the theorems of Napoleon and Morley as special cases. The remaining papers in this series deal with complex triangle coordinates and the triangle functions of special points. A useful background for all three is [5], which catalogues and references many properties of triangles and their special points. The necessary background for complex geometry is outlined in the next section. Manuscript received Cross ratios in the complex planeWe give here only the basic mathematical prerequisites; proofs and further details may be found in [20], [22] or in many complex analysis texts.Identify the Euclidean plane with the complex numbers C, and adjoin the "point at infinity" ~: C~ ,=Cw{~}. The cross ratio of distinct points a, b, c and d in Ca is the number

Triangles II: Complex triangle coordinates

“…(Since from MTST, the Miquel triangles of the symmedian point then have shape ( A -2)/(2A -1), this justifies the claim of Example 6.3 of [4]. )…”

“…Two triangles are similar whenever they have the same shape and anti-similar whenever they have conjugate shapes. The angles of a triangle and its shape thus determine each other: for A := & bac, B,= zk cba and C := ~ acb, we have [4] 1 -e -era Proof. Written out in full, the mapping is We then have that ZAZ~,ZA,, : --1.…”

“…The first paper of the series [4] developed the notion of triangle shapes. We now use cross ratios to coordinatize the complex plane relative to a given triangle.…”

Propellers in Affine Cayley–Klein Planes

“…The lemma below shows that the shape of any triangle characterizes this triangle up to similarity. Also it should be noticed that in the same spirit the classes of similar triangles are treated in [13] (for the Euclidean plane), in [16] (for the isotropic plane), and in [1] (for finite planes). Proof.…”

Convolution filters for polygons and the Petr–Douglas–Neumann theorem

“…Napoleon's theorem for equilateral triangles presented here has already been very successful for analyzing triangle transformations (Nicollier and Stadler 2013; Nicollier 2013): we generalize in particular the powerful notion of shape of triangles by defining the shape of an arbitrary planar polygon as the homogeneous coordinates given by the spectrum of this polygon without offset. See Lester (1996aLester ( ,b, 1997 for another shape of triangles.…”