## X(92) (CEVAPOINT OF INCENTER AND CLAWSON 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           csc 2A : csc 2B : csc 2C
Barycentrics    sec A : sec B : sec C

Let LA be the line through X(4) parallel to the internal bisector of angle A, and let
A' = BC∩LA. Define B' and C' cyclically.

Alexei Myakishev, "The M-Configuration of a Triangle," Forum Geometricorum 3 (2003) 135-144,

proves that the lines AA', BB', CC' concur in X(92). He notes that another construction follows from Proposition 2 of the article: let A1 be the midpoint of the arc BC of the circumcircle that passes through A, and let A2 be the point, other than A, in which the A-altitude meets the circumcircle. Let A" = A1A2∩BC. Define B" and C" cyclically. Then the lines AA", BB", CC" concur in X(92).

X(92) lies on these lines:
1,29    2,273    4,8    7,189    19,27    25,242    31,162    38,240    40,412    47,91    55,243    57,653    85,331    100,917    226,342    239,607    255,1087    257,297    264,306    304,561    406,1068    608,894

X(92) = isogonal conjugate of X(48)
X(92) = isotomic conjugate of X(63)
X(92) = anticomplement of X(1214)
X(92) = X(I)-Ceva conjugate of X(J) for these (I,J): (85,342), (264,318), (286,4), (331,273)
X(92) = cevapoint of X(I) and X(J) for these (I,J): (1,19), (4,281), (47,48), (196,278)
X(92) = X(I)-cross conjugate of X(J) for these (I,J): (1,75), (4,273), (19,158), (48,91), (226,2), (281,318)
X(92) = crosspoint of X(I) and X(J) for these (I,J): (85,309), (264,331)
X(92) = crossdifference of any two points on line X(810)X(822)
X(92) = X(275)-aleph conjugate of X(47)
X(92) = X(I)-beth conjugate of X(J) for these (I,J): (92,278), (312,329), (648,57)

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