## X(51) (CENTROID OF ORTHIC TRIANGLE)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
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You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

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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           a2cos(B - C) : b2cos(C - A) : c2cos(A - B)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a2(b2 + c2) - (b2 - c2)2]

Barycentrics    a3cos(B - C) : b3cos(C - A) : c3cos(A - B)

X(51) lies on these lines:
2,262    4,185    5,52    6,25    21,970    22,182    23,575    24,578    26,569    31,181    39,237    44,209    54,288    107,275    125,132    129,137    130,138    199,572    210,374    216,418    381,568    397,462    398,463    573,1011

X(51) is the {X(5),X(143)}-harmonic conjugate of X(52).

X(51) = reflection of X(210) in X(375)
X(51) = isogonal conjugate of X(95)
X(51) = complement of X(2979)
X(51) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,53), (5,216), (6,217)
X(51) = X(217)-cross conjugate of X(216)
X(51) = crosspoint of X(I) and X(J) for these (I,J): (4,6), (5,53)
X(51) = crosssum of X(I) and X(J) for these (I,J): (2,3), (6,160), (54,97)
X(51) = crossdifference of any two points on line X(323)X(401)

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