On Converses of Napoleon’s Theorem and a Modified Shape Function

The (negative) Torricelli triangle T1(ABC) of a nondegenerate (positively oriented) triangle ABC is defined to be the triangle A1B1C1, where ABC1, BCA1, and CAB1 are the equilateral triangles drawn outwardly on the sides of ABC. It is known that not every triangle is the Torricelli triangle of some initial triangle, and triangles that are not Torricelli triangles are characterized in [28]. In the present article it is shown that, by extending the definition of T1 such that degenerate triangles are included, the mapping T1 becomes bijective and every triangle is then the Torricelli triangle of a unique triangle. It is also shown that T1 has the smoothing property, i.e., that the process of iterating the operations T1 converges, in shape, to an equilateral triangle for any initial triangle. Analogous statements are obtained for internally erected equilateral triangles, and the proofs give rise to a slightly modified form of June Lester’s shape function which is expected to be useful also in other contexts. Several further results pertaining to the various triangles that arise from the configuration created by ABC1, BCA1, and CAB1 are derived. These refer to Brocard angles, perspectivity properties, and (oriented) areas. ∗The first named author was supported by a research grant from Yarmouk University 0138-4821/93 $ 2.50 c © 2006 Heldermann Verlag 364 M. Hajja et al.: On Converses of Napoleon’s Theorem. . .