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.
Similar content being viewed by others
References
Barotti, A.,Una proprietà degli n-agoni che si ottengono transformando in una affinità un n-agono regolare. Boll Un. Mat. Ital (3)10 (1955), 96–98.
Hodgson, J. E.,Orthocentric properties of the plane directed n-line. Trans. Amer. Math. Soc.13 (1912), 199–231.
Johnson, R. A.,Modern geometry. Houghton Mifflin, New York, 1929.
Kapteyn, W.,Over de merkwaardige punten van der driehoek. Verh. Konink. Acad. Wetensch. Belgiē (1) III, Nr. 3 (1895), 3–33.
Kimberling, C.,Central points and central lines in the plane of a triangle. Math. Mag.67 (1994), 163–187.
Jung, H. E. W.,Über die Lage der Hauptträgheitsachsen von Punktsystem in der Ebene. Arch. Math. Phys. (3)13 (1908), 285–286.
Kürschák, J.,Anwendung der komplexen Zahlen zum Beweise eines elementargeometrisches Satzes. Arch. Math. Phys. (3)8 (1905), 285–286.
Lester, J. A.,Points of difference: Relative infinity in the Euclidean plane. Mitt. Math. Ges. Hamburg13 (1993), 93–117.
Lester, J. A.,Triangles II: Complex triangle coordinates. (to appear in Aequationes Math.).
Lester, J. A.,Triangles III: Complex triangle functions. (to appear in Aequationes Math.).
Lester, J. A.,A generalization of Napoleon's theorem. C.R. Math. Rep. Acad. Sci. Canada16 (1994), 253–257.
Meyer, W. Fr.,Über den Ptolemäischen Satz. Arch. Math. Phys. (3)7 (1904), 1–15.
Meyers L. F.,Solution to problem 464. Crux. Math.6 (1980), 185–187.
Morley, F. andMorley, F. V.,Inversive geometry. Chelsea, New York, 1954.
Neumann, B. H.,Some remarks on polygons. J. London Math. Soc.16 (1941), 230–245.
Rigby, J.,Napoleon revisited. J. Geom.33 (1988), 129–146.
Samaga, H.-J.,A unified approach to Miquel's theorem and its degenerations. InGeometry and differential geometry. [Lecture Notes in Mathematics, No. 792]. Springer, Berlin, 1980, pp. 132–142.
Schaeffer, H. andBenz, W.,Peczar-Doppelverhältnisidentitäten zum allgemeinen Satz von Miquel. Abh. Math. Sem. Univ. Hamburg42 (1974), 228–235.
Shick, J.,Beziehung zwischen Isogonalcentrik und Invariantentheorie. Bayer. Akad. Wiss. Sitzungsber.30 (1900), 249–272.
Schwerdtfeger, H.,The geometry of complex numbers. Dover, New York, 1979.
Wells, D.,The Penguin dictionary of curious and interesting geometry. Penguin Books, London, 1991.
Yaglom, I. M.,Complex numbers in geometry. Academic Press, New York, 1968.
Zwikker, C.,The advanced geometry of plane curves and their applications. Dover, New York, 1963.