Circles in Morley's Triangles: What is this about?
A Mathematical Droodle
What if applet does not run? |
(The numbers are clickable and so may be modified.)
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' O_{A}, O_{B} and O_{C}, and those of triangles BCA', CAB', and ABC' Q_{A}, Q_{B} and Q_{C}, respectively. Let angles
Since it's known that
Similar statements are true with regard to O_{B}Q_{B} and O_{C}Q_{C}, 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 Q_{A} is the circumcenter of ΔBCA', ΔBA'Q_{A} is isosceles, so that
∠Q_{A}A'B = (180^{0} - 2z)/2 = 90^{0} - z, |
as claimed.
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Copyright © 1996-2018 Alexander Bogomolny