Cutting circles and the Morley theorem (Q1356597)

A circle \(k\) is called a 1-cutting circle (``1-Schnittkreis'') of a triangle ABC iff this circle \(k\) cuts out of the lines \(BC\), \(CA\), \(AB\) a chord of length \(\overline {BC}\), \(\overline {CA}\), \(\overline {AB}\), respectively. In Elem. Math. 47, No. 4, 158-168 (1992; Zbl 0777.51007) the author proved: For each triangle \(ABC\) (in the Euclidean plane \(\mathbb{R}^2)\) there exists, additionally to the circumscribed circle \(c\), three and only three 1-cutting circles; their midpoints are the corners of an equilateral triangle \(UVW\) which is circumscribed about \(c\); in general, these points \(U,V,W\) are not constructible, using lines and circles only. The author now proves the Theorem: For each triangle \(ABC\) the triangle \(UVW\) of the midpoints of 1-cutting circles and the Morley triangle \(XYZ\) have pairwise parallel sides with equal orientation.