## X(21) (SCHIFFLER POINT)

 Interactive Applet

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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           1/(cos B + cos C) : 1/(cos C + cos A) : 1/(cos A + cos B)
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/(b + c)

Barycentrics    a/(cos B + cos C) : b/(cos C + cos A) : c/(cos A + cos B)

Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X(21).

Lev Emelyanov and Tatiana Emelyanova, A note on the Schiffler point, Forum Geometricorum 3 (2003) pages 113-116.

The name of this point honors Kurt Schiffler.

X(21) is the {X(2),X(3)}-harmonic conjugate of X(404).

X(21) = midpoint of X(1) and X(191)
X(21) = isogonal conjugate of X(65)
X(21) = isotomic conjugate of X(1441)
X(21) = inverse-in-circumcircle of X(1325)
X(21) = anticomplement of X(442)
X(21) = X(I)-Ceva conjugate of X(J) for these (I,J): (86,81), (261,333)
X(21) = cevapoint of X(I) and X(J) for these (I,J): (1,3), (9,55)

X(21) = X(I)-cross conjugate of X(J) for these (I,J):
(1,29), (3,283), (9,333), (55,284), (58,285), (284,81), (522,100)

X(21) = crosspoint of X(86) and X(333)
X(21) = crosssum of X(I) and X(J) for these (I,J): (1,1046), (42,1400), (1254,1425), (1402,1409)
X(21) = crossdifference of any two points on line X(647)X(661)
X(21) = X(I)-Hirst inverse of X(J) for these (I,J): (2,448), (3,416), (4,425)
X(21) = X(I)-beth conjugate of X(J) for these (I,J): (21,58), (99,21), (643,21), (1043,1043), (1098,21)

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