## X(554) (INTERSECTION OF LINES X(1)X(30) AND X(14)X(226))

 Interactive Applet

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 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           sec(A/2) csc(A/2 + π/3) : sec(B/2) csc(B/2 + π/3) : sec(C/2) csc(C/2 + π/3)
Barycentrics    sin A sec(A/2) csc(A/2 + π/3) : sin B sec(B/2) csc(B/2 + π/3) : sin C sec(C/2) csc(C/2 + π/3)

Suppose X and Y are triangle centers. Let

YA = (Y of the triangle XBC),
YB = (Y of the triangle XCA),
YC = (Y of the triangle XAB).

Let A' = (XYA intersect BC), and define B' and C' cyclically. In

Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438,

Question A is this: for what choices of X and Y do the lines AA', BB', CC' concur? A solution (X,Y) will here be called the (X,Y)-answer to Question A. X(554) is the (X(1),X(13))-answer to Question A. (In the reference, see (9) on page 435, with Y = X(13).)

X(554) lies on these lines: 1,30    7,1082    14,226    75,299

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