Grégoire Nicollier, Two six-circle theorems for cyclic pentagons,
Forum Geometricorum, 16 (2016) 347--354.
Abstract. Miquel's pentagram theorem is true for any pentagon. We consider the pentagram obtained by producing the sides of a pentagon and prove two further six-circle theorems, the first for a cyclic pentagram and the second for a cyclic pentagon. If the pentagram is cyclic, consecutive circumcircles of the ear edges issued from the same pentagon vertex have concyclic alternate intersections. If the pentagon is cyclic, alternate intersections of the circumcircles of the rooted ears issued from the same pentagon vertex are concyclic (a rooted ear is an ear extended by the neighboring sides of the pentagon). Among related results, we also show that the circumcircle of an ear producing opposite sides of a cyclic quadrilateral and the circumcircle of the corresponding rooted ear are both tangent to the same two circles centered at the circumcenter of the quadrilateral.
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