Triangles. I: Shapes (Q1924288)

This is the first of a series of three papers that examine Euclidean triangle geometry via complex cross-ratios [as developed, for example, by \textit{I. M. Yaglom} in his book `Complex numbers in geometry', Academic Press, New York (1966; Zbl 0147.20201)]. In this first paper, the author examines analytic methods for proving similarity theorems. She observes that, by using cross-ratios, any Euclidean triangle may be classified up to similarity by a single complex number, which she calls the shape of the triangle. In particular, she discusses the shapes of Miquel triangles. Finally, she examines two general shape theorems, the second of which has the theorems of Napoleon and Morley as special cases.