## X(903) (X(1)-BLAIKIE TRANSFORM OF X(2))

 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            bc/(b + c - 2a) : ca/(c + a - 2b) : ab/(a + b - 2c)
Barycentrics    1/(b + c - 2a) : 1/(c + a - 2b) : 1/(a + b - 2c) (Darij Grinberg, 12/28/02)

James Blaikie (1847-1929) proposed the following problem. Let O be any point in the plane of triangle ABC, and let any straight line g through O meet BC in P, CA in Q, AB in R; then, if points P', Q', R' be taken on the line so that

PO = OP',       QO = OQ',       RO = OR',

prove that AP', BQ', CR' are concurrent.

Darij Grinberg introduces the term Blaikie point of O and g for the point Z of concurrence. If

O = x : y : z and g = [k : l : m] (barycentric coordinates),

then Z has first barycentric 1/[k(y-z) - (ly-mz)]. Given a point S = u : v : w, Grinberg then defines the S-Blaikie transform of O as the Blaikie point of O and OS. The first barycentric of Z can be written as

1/[yw(y+x) + zv(z+x) - yz(2u+v+w)].

Visit Blaikie theorem in barycentrics. (Darij Grinberg, 12/28/02)

X(903) lies on these lines:
2,45    7,528    27,648    75,537    86,99    310,670    320,519    335,536    350,889    527,666    675,901    812,1022

X(903) = reflection of X(I) in X(J) for these (I,J): (2,1086), (3,190)
X(903) = isogonal conjugate of X(902)
X(903) = isotomic conjugate of X(519)
X(903) = X(I)-cross conjugate of X(J) for these (I,J): (320,86), (519,2)

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