## X(1986) (HATZIPOLAKIS REFLECTION POINT)

 Interactive Applet

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 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) = (1 + cos 2B + cos 2C) sin 3A csc 2A
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A'B'C' be the orthic triangle of triangle ABC. Let AB be the reflection of A in C', and define AC, BC, BA, CA, CB functionally. Then the nine-point circles of the triangles

AABAC,      BBCBA,      CCACB,

concur in X(1986). (Antreas Hatzipolakis, Hyacinthos 7868, 9/12/03; coordinates by Barry Wolk, Hyacinthos 7876, 9/13/03)

X(1986) lies on these lines:
9,94    6,74    24,110    25,399    113,403    125,389    186,323    542,1843    648,1300    1844,1845

X(1986) = reflection of X(I) in X(J) for these (I,J): (4,1112), (74,974), (125,389)
X(1986) = X(4)-Ceva conjugate of X(403)
X(1986) = crosspoint of X(4) and X(186)
X(1986) = crosssum of X(3) and X(265)

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