## X(37) (CROSSPOINT OF INCENTER AND CENTROID)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

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           b + c : c + a : a + b
Barycentrics    a(b + c) : b(c + a) : c(a + b)

Let A'B'C' be the cevian triangle of X(1). Let A" be the centroid of triangle AB'C', and define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(37). (Eric Danneels, Hyacinthos 7892, 9/13/03)

X(37) lies on these lines:
1,6    2,75    3,975    7,241    8,941    10,594    12,225    19,25    21,172    35,267    38,354    39,596    12,225    41,584    48,205    63,940    65,71    73,836    78,965    82,251    86,190    91,498    100,111    101,284    141,742    142,1086    145,391    158,281    171,846    226,440    256,694    347,948    513,876    517,573    537,551    579,942    626,746    665,900    971,991

X(37) is the {X(1),X(9)}-harmonic conjugate of X(6).

X(37) = midpoint of X(I) and X(J) for these (I,J): (75,192), (190,335)
X(37) = isogonal conjugate of X(81)
X(37) = isotomic conjugate of X(274)
X(37) = complement of X(75)

X(37) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,42), (2,10), (4,209), (9,71), (10,210), (190,513), (226,65), (321,72), (335,518)

X(37) = cevapoint of X(213) and X(228)
X(37) = X(I)-cross conjugate of X(J) for these (I,J): (42,65), (228,72)
X(37) = crosspoint of X(I) and X(J) for these (I,J): (1,2), (9,281), (10,226)
X(37) = X(1)-line conjugate of X(238)
X(37) = crosssum of X(I) and X(J) for these (I,J): (1,6), (57,222), (58,284), (1333,1437)
X(37) = crossdifference of any two points on line X(36)X(238)
X(37) = X(10)-Hirst inverse of X(740)
X(37) = X(1)-aleph conjugate of X(1051)
X(37) = X(I)-beth conjugate of X(J) for these (I,J): (9,37), (644,37), (645,894), (646,37), (1018,37)

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