Thebault circles of the triangle in an isotropic plane

Abstract. In this paper the existence of three circles, which touch the circumscribed circle and Euler circle of an allowable triangle in an isotropic plane, is proved. Some relations between these three circles and elements of a triangle are investigated. Formulae for their radii are also given. AMS subject classifications: 51N25 Key words: isotropic plane, Thebault’s circles In [6] V. Thébault considered the following figure in Euclidean geometry. Let ABC be any triangle and Am, Bm, Cm the midpoints of its sides BC, CA, AB. If the medians AAm, BBm, CCm of that triangle meet its Euler circle Ke = AmBmCm (or the nine–point circle, i.e., the circle through the feet of the three altitudes, the midpoints of the three sides, and the midpoints of the segments from the three vertices to the orthocenter [2]) again at the points L, M, N, then there exist three circles D, E, F, which touch the circumscribed circle Ko = ABC of the triangle ABC at the points A, B, C, and they touch the Euler circle Ke successively at the points L, M, N. Radii of these circles are d, e, f, respectively, where for example d =