## X(1594) (RIGBY-LALESCU ORTHOPOLE)

 Interactive Applet

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec A + 2 cos(B - C)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos(2B-C) cos(A-C) + cos(2C-B) cos(A-B)
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Traian Lalescu (Trajan Lalesco) proved in "A Class of Remarcable Triangles," Gazeta Matematica 20 (1915) 213 [in Romanian], that if triangles DEF and D'E'F' are inscribed in a circle and directed arclengths satisfy

arc DD' + arc EE' + arc FF' = 0 mod 2π,

then the Simson lines of D,E,F with respect to D',E',F' and the Simson lines of D',E',F' with respect to D,E,F concur in the midpoint X of the segment of the orthocenters of DEF and D'E'F'. Daniel Vacaretu considered triangles DEF and D'E'F' associated with left and right isoscelizers and inscribed in the sine-triple-angle circle. He obtained the second set of trilinears shown above for the midpoint X. (See also the bicentric pair PU(61).)

In Episodes in Nineteenth and Twentieth Century Euclidean Geometry,, page 132, Ross Honsberger presents X(1594) as the orthopole of the six sides of two triangles and as the point common to six Simson lines. Honsberger calls this orthopole the Rigby Point. (Notes on Lalescu and Honsberger received from D. Vacaretu, 19/16/03)

X(1594) lies on these lines:
2,3    6,70    50,252    53,566    67,1173    96,275    125,389    128,136    232,1508    264,847    325,1235    933,1166    1209,1216    1225,1238

X(1594) = inverse-in-nine-point-circle of X(186)
X(1594) = inverse-in-orthocentroidal-circle of X(24)
X(1594) = X(933)-Ceva conjugate of X(523)
X(1594) = crosspoint of X(I) and X(J) for these (I,J): (4,93), (264,275)
X(1594) = crosssum of X(I) and X(J) for these (I,J): (3,49), (184,216)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.