Summary
This 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 coordinatize the Euclidean plane using cross ratios, and use these triangle coordinates to prove theorems about triangles. We develop a complex version of Ceva's theorem, and apply it to proofs of several new theorems. The remaining paper of this series will deal with complex triangle functions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Gallatly, W.,The modern geometry of the triangle. 2nd edition, Hodgson, London, 1913.
Gale, C.,From Euclid to Descartes to Mathematica to oblivion? Mathematical entertainments (column) in Math. Intelligencer14 (1992), no. 2, 68–71.
Kimberling, C.,Central points and central lines in the plane of a triangle. Math. Mag.67 (1994), 163–187.
Lester, J. A.,Triangles I: Shapes. Aequationes Math.52 (1996), 30–54.
Lester, J. A.,Trianigles III: Complex triangle functions. To appear in Aequationes Math.
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 (Proc. Conf. Haifa, 1979) [Lecture Notes in Math., No. 792]. Springer, Berlin, 1980, pp. 132–142.
Schaeffer, H. andBenz, W.,Peczar-Doppelverhältnisidentitäten zum allgemeinen Satz von Miquel. Abh. Math. Sem. 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.
Yaglom, I. M. Complex numbers in geometry. Academic Press, New York, 1968.