One of the more enduring results of the resurgence of interest in Euclidean geometry during the 19th century is the Kiepert hyperbola attached to any triangle.¹ Given a triangle $\Delta ABC$ and an acute angle $\phi$, construct isosceles triangles $\Delta BCX$, $\Delta CAY$ and $\Delta ABZ$ with base angle $\phi$ on the sides of $\Delta ABC$. Then the lines $\overline{XA}$, $\overline{YB}$ and $\overline{ZC}$, connecting a vertex of the original triangle to the apex of the opposite isosceles triangle, are concurrent. This result generalizes the concurrence of the medians (limiting case $\phi \rightarrow 0$), the Fermat point ($\phi = 60^{\circ}$), the "Napoleon points" ($\phi = \pm 30^{\circ}$) and the altitudes (limiting case $\phi \rightarrow 90^{\circ}$), plus several other less well-known cases. Moreover, as the base angle $\phi$ varies, the points of concurrency move along a hyperbola, now known as the Kiepert hyperbola of $\Delta ABC$.

Interest also attaches to the "Kiepert triangles" $\Delta XYZ$ formed by the apexes of the three isosceles triangles,² and their relation to the original triangle. The limiting case $\phi = 0$ gives the medial triangle, which is classically known to be similar to the original triangle. On the other hand, another well-known 19th century result, usually called ``Napoleon's Theorem", says that when $\phi = 30^{\circ}$, the Kiepert triangle is always equilateral, regardless of the original $\Delta ABC$. This talk will discuss the geometry of the Kiepert triangles, and will show how it can be illuminated by a one-parameter group of Möbius transformations of the complex plane. This line of thought was suggested by Andrew Miller.

1. Eddy, R. H. and Fritsch, R., "The Conics of Ludwig Kiepert: A Comprehensive Lesson in the Geometry of the Triangle," Math. Mag. 67 (1994), 188-205.
2. van Lamoen, F. and You, P., "The Kiepert Pencil of Kiepert Hyperbolas," Forum Geometricorum 1 (2001), 125 - 132.