## X(54) (KOSNITA POINT)

 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           sec(B - C) : sec(C - A) : sec(A - B)
Barycentrics    sin A sec(B - C) : sin B sec(C - A) : sin C sec(A - B)

John Rigby, "Brief notes on some forgotten geometrical theorems," Mathematics and Informatics Quarterly 7 (1997) 156-158.

Let O be the circumcenter of triangle ABC, and Oa the circumcenter of triangle BOC. Define Ob and Oc cyclically. Then the lines AOa, BOb, COc concur in X(54). For details and generalization, see

Darij Grinberg, A New Circumcenter Question

X(54) lies on these lines:
2,68    3,97    4,184    5,49    6,24    12,215    36,73    39,248    51,288    64,378    69,95    71,572    72,1006    74,185    112,217    140,252    156,381    186,389    276,290    575,895    826,879

X(54) is the {X(5),X(49)}-harmonic conjugate of X(110).

X(54) = midpoint of X(3) and X(195)
X(54) = reflection of X(195) in X(1493)
X(54) = isogonal conjugate of X(5)
X(54) = isotomic conjugate of X(311)
X(54) = inverse-in-circumcircle of X(1157)
X(54) = complement of X(1210)
X(54) = anticomplement of X(1209)
X(54) = X(I)-Ceva conjugate of X(J) for these (I,J): (95,97), (288,6)
X(54) = cevapoint of X(6) and X(184)
X(54) = X(I)-cross conjugate of X(J) for these (I,J): (3,96), (6,275), (186,74), (389,4), (523,110)
X(54) = crosspoint of X(95) and X(275)
X(54) = crosssum of X(I) and X(J) for these (I,J): (3,195), (51,216), (627,628)

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