## X(2092) (DANNEELS-APOLLONIUS PERSPECTOR)

 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            f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)2 + a(b + c)(b2 + c2)
Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A' be the point common to Apollonius circle and the A-excircle, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the cevian triangle of X(6), and the perspector is X(2092). (Eric Danneels, Hyacinthos, #8070, 10/01/03). (Triangle A'B'C' is also perspective to the cevian triangle of X(1); see X(1682).)

X(2092) lies on these lines: 2,314    3,6    9,43    10,37    42,181    69,980    71,213    115,119    214,1015    232,1172    407,1880    442,1738    1193,1682

X(2092) = midpoint of X(I) and X(J) for these (I,J): (256,1045), (2019,2020)
X(2092) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,960), (100,512), (1415,647)
X(2092) = crosspoint of X(I) and X(J) for these (I,J): (2,65), (6,37), (429,1211)
X(2092) = crosssum of X(I) and X(J) for these (I,J): (2,81), (6,21), (58,572), (1169,1798)

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