Cutting Circles and the Morley Theorem

Abstract: For each triangle $ABC$ exist, additionally to the circumscribed circle $c$, three and only three circles with the following property: The circle cuts out of the lines ${\rm g}(BC)$, ${\rm g}(CA), {\rm g}(AB)$ chords of the length $BC, CA, AB$ respectively. Their midpoints are the corners of an equilateral triangle, which is circumscribed about $c$ and has parallel sides to the Morley triangle. The proof is elementary (with a trigonometrical calculation) and uses a property, also using for a known proof of the Morley theorem.

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