X(17) (1ST NAPOLEON 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           csc(A + π/6) : csc(B + π/6) : csc(C + π/6)
= sec(A - π/3) : sec(B - π/3) : sec(C - π/3)

Barycentrics    a csc(A + π/6) : b csc(B + π/6) : c csc(C + π/6)

Let X,Y,Z be the centers of the equilateral triangles in the construction of X(13). The lines AX, BY, CZ concur in X(17).

John Rigby, "Napoleon revisited," Journal of Geometry, 33 (1988) 126-146.

X(17) lies on these lines:
2,62    3,13    4,15    5,14    6,18    12,203    16,140    76,303    83,624    202,499    275,471    299,635    623,633

X(17) is the {X(231),X(1209)}-harmonic conjugate of X(18).

X(17) = reflection of X(627) in X(629)
X(17) = isogonal conjugate of X(61)
X(17) = isotomic conjugate of X(302)
X(17) = complement of X(627)
X(17) = anticomplement of X(629)
X(17) = X(I)-cross conjugate of X(J) for these (I,J): (16,14), (140,18), (397,4)

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