# Triangles I: Shapes

@article{Lester1996TrianglesIS, title={Triangles I: Shapes}, author={June A. Lester}, journal={aequationes mathematicae}, year={1996}, volume={52}, pages={30-54} }

SummaryThis 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.

#### 28 Citations

Triangles III: Complex triangle functions

- Mathematics
- 1997

SummaryThis paper is the third in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first two papers, we looked at triangle shapes and triangle coordinates. In… Expand

Triangles II: Complex triangle coordinates

- Mathematics
- 1996

SummaryThis paper is the second in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first paper of the series, we examined triangle shapes. In this paper, we… Expand

A Note on Similar-Perspective Triangles

- Mathematics
- 2006

An old theorem of F. E. Wood (9) states that if two triangles in the Euclidean plane are directly similar and perspective from a point then either their sides are parallel in pairs or their… Expand

Complex Bézier curves and the geometry of polygons

- Mathematics, Computer Science
- Comput. Aided Geom. Des.
- 2010

Several well-known theorems on polygons such as the Napoleon-Barlotti Theorem, the Petr-Douglas-Neumann Theorem and the Fundamental Decomposition Theorem of polygons to regular polygons are proved. Expand

Complex Bézier Curves and the Geometry of Polynomials

- Mathematics, Computer Science
- Curves and Surfaces
- 2010

It is shown that the location of the complex roots of the polynomial dictates geometrical constraints on the shape of the control polygon of a complex Bezier curve over a complex interval. Expand

Shapes of tetrahedra

- Mathematics
- 2002

Abstract. The equivalence classes of triangles and tetrahedra with respect to the
group of the space dilatations and translations can be expressed by
quaternions and ordered pairs of quaternions,… Expand

A Characterization of the Centroid Using June Lester's Shape Function

- Mathematics
- 2006

The notion of triangle shape is used to give another proof of the fact that if P is a point inside triangle ABC and if the cevian triangle of P is similar to ABC in the natural order, then P is the… Expand

Shape-regular polygons in finite planes

- Mathematics
- 1996

The notion of shape in the Gaussian plane was introduced by Lester [5] and extended by Artzy [1]. In this paper we generalize this notion in the affine planesAG(2,q) over the Galois fieldGF(q), q=pr… Expand

Curves on the shape sphere

- Mathematics
- 2003

Using a special conformai map between the two-dimensional sphere and the extended plane, we describe some classes of curves on the sphere. We also discuss a differential geometric invariant… Expand

Degree of Triangle Centers and a Generalization of the Euler Line

- Mathematics
- 2010

We introduce a concept "degree of triangle centers", and give a formula expressing the degree of triangle centers on generalized Euler lines. This generalizes the well known 2 : 1 point configuration… Expand

#### References

SHOWING 1-10 OF 11 REFERENCES

Triangles III: Complex triangle functions

- Mathematics
- 1997

SummaryThis paper is the third in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first two papers, we looked at triangle shapes and triangle coordinates. In… Expand

Triangles II: Complex triangle coordinates

- Mathematics
- 1996

SummaryThis paper is the second in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first paper of the series, we examined triangle shapes. In this paper, we… Expand

Central points and central lines in the plane of a triangle

- Mathematics
- 1994

Triangle geometry ranks among the most enduring topics in all of mathematics. A treasury of triangle lore abounds in Euclid's Elements of 2.3 millenia ago, and still today interesting elementary… Expand

The Penguin Dictionary of Curious and Interesting Geometry

- Computer Science, History
- 1991

What do the Apollonian gasket, Dandelin spheres, interlocking polyominoes, Poncelet's porism, Fermat points, Fatou dust, the Vodernberg tessellation, the Euler line and the unilluminable room have in… Expand