X(55) (INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE))

 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           a(b + c - a) : b(c + a - b) : c(a + b - c)
= 1 + cos A : 1 + cos B : 1 + cos C
= cos2(A/2) : cos2(B/2) : cos2(B/2)
= tan(B/2) + tan(C/2) : tan(C/2) + tan(A/2) : tan(A/2) + tan(B/2)

Barycentrics    a2(b + c - a) : b2(c + a - b) : c2(a + b - c)

X(55) = center of homothety of three triangles:    tangential, intangents, and extangents. Also, X(55) is the pole-with-respect-to-the-circumcircle of the trilinear polar of X(1). These propreties and others are given in

O. Bottema and J. T. Groenman, "De gemene raaklijnen van de vier raakcirkels van een driehoek, twee aan twee," Nieuw Tijdschrift voor Wiskunde 67 (1979-80) 177-182.

X(55) is the {X(1),X(3)}-harmonic conjugate of X(56).

X(55) = reflection of X(1478) in X(495)
X(55) = isogonal conjugate of X(7)
X(55) = inverse-in-circumcircle of X(1155)

X(55) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,6), (3,198), (7,218), (8,219), (9,220), (21,9), (59,101), (104,44), (260,259), (284,41)

X(55) = cevapoint of X(42) and X(228) for these (I,J)
X(55) = X(I)-cross conjugate of X(J) for these (I,J): (41,6), (42,33), (228,212)
X(55) = crosspoint of X(I) and X(J) for these (I,J): (1,9), (3,268), (7,277), (8,281), (21,284), (59,101)

X(55) = crosssum of X(I) and X(J) for these (I,J): (1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234), (241,1362), (513,1086), (905,1364), (1361,1465)

X(55) = crossdifference of any two points on line X(241)X(514)
X(55) = X(I)-Hirst inverse of X(J) for these (I,J): (6,672), (43,241)
X(55) = X(1)-line conjugate of X(241)
X(55) = X(I)-beth conjugate of X(J) for these (I,J): (21,999), (55,31), (100,55), (200,200), (643,2), (1021,1024)

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