## X(399) (PARRY 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) = 5 cos A - 4 cos B cos C - 8 sin B sin C cos2A

Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,B,A)

Let L, M, N be lines through A, B, C, respectively, parallel to the Euler line. Let L' be the reflection of L in sideline BC, let M' be the reflection of M in sideline CA, and let N' be the reflection of N in sideline AB. The lines L', M', N' concur in X(399).

Cyril Parry, Problem 10637, American Mathematical Monthly 105 (1998) 68.

X(399) lies on these lines:
3,74    4,195    6,13    30,146    155,382    394,541

X(399) = isogonal conjugate of X(1138)
X(399) = reflection of X(I) in X(J) for these (I,J): (3,110), (74,1511), (265,113)
X(399) = X(I)-Ceva conjugate of X(J) for these (I,J): (30,3), (323,6)

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