X(17) = 1st NAPOLEON POINT

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

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

Notes

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).

Dao Thanh Oai, Equilateral Triangles and Kiepert Perspectors in Complex Numbers, Forum Geometricorum 15 (2015) 105-114.

Dao Thanh Oai, A family of Napoleon triangles associated with the Kiepert configuration, The Mathematical Gazette 99 (March 2015) 151-153.

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

If you have The Geometer’s Sketchpad, you can view 1st Napoleon point. If you have GeoGebra, you can view 1st Napoleon point.

X(17) lies on the Napoleon cubic and 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),:ref:X(1209) <X(1209)>}-harmonic conjugate of X(18). For a list of other harmonic conjugates of X(17), click Tables at the top of this page.

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) = circumcircle-inverse of X(32627)

X(17) = X(i)-cross conjugate of X(j) for these (i,j): (16,14), (140,18), (397,4)

X(17) = polar conjugate of X(473)

X(17) = trilinear product of vertices of outer Napoleon triangle

X(17) = Kosnita(X(13),:ref:X(3) <X(3)>) point

X(17) = Kosnita(X(17),:ref:X(17) <X(17)>) point

X(17) = Cundy-Parry Phi transform of X(13)

X(17) = Cundy-Parry Psi transform of X(15)

X(17) = trilinear pole of line X(523)X(14446)

X(17) = X(63)-isoconjugate of X(10642)