## X(1487) (NAPOLEON CEVAPOINT)

 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           f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/[(3 - cot B cot C)(3 - cot2A)]
Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let N denote the nine-point center, X(5). Let NA = N-of-triangle NBC, and define NB and NC cyclically. Triangle NANBNC is perspective to ABC, and X(1487) is the perspector. X(1487) is the cevapoint of the Napoleon points, X(17) and X(18). (Coordinates found by Paul Yiu.)

The construction just given for X(1487) shows that it is a solution X of the "four-triangle problem" posed in

C. Kimberling, "Triangle centers as functions," Rocky Mountain Journal of Mathematics 23 (1993) 1269-1286. See Section 5; a complete solution to the problem remains to be found.

X(1487) lies on these lines: 4,252    5,1173    140,930

X(1487) = isogonal conjugate of X(1493)
X(1487) = cevapoint of X(17) and X(18)
X(1487) = X(523)-cross conjugate of X(930)

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