Circles in Morley's Triangles: What is this about?
A Mathematical Droodle
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Explanation. See also the page on Morley Constellation.

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Copyright © 1996-2018 Alexander Bogomolny
What's happening is this. Let r be a number between 0 and 1. We shall perform a construction similar to but more general than Morley's. For vertex A, measure angles rA from both sides, and similarly for the vertices B and C. The lines adjacent to one of the sides of the given triangle intersect forming seven triangles: the middle one -- Morley's if

Theorem (by Nikos Dergiades with an important correction by Samuele Mongodi)
Let A'B'C' denote the Morley triangle. Denote the circumcenters of triangles AB'C', BC'A', and CA'B' OA, OB and OC, and those of triangles BCA', CAB', and ABC' QA, QB and QC, respectively. Let angles
Since it's known that
Similar statements are true with regard to OBQB and OCQC, which means that the three lines meet at the center of Morley's triangle.
(Some of the triangles have already been considered elsewhere.)

Samuele Mongodi found a flaw in the original Nikos' argument and offered the following fix:
Since QA is the circumcenter of ΔBCA', ΔBA'QA is isosceles, so that
∠QAA'B = (1800 - 2z)/2 = 900 - z, |
as claimed.

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Copyright © 1996-2018 Alexander Bogomolny
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