X(42) = X(42) CROSSPOINT OF INCENTER AND SYMMEDIAN POINT¶

Trilinears

\(a(b + c) : b(c + a) : c(a + b)\)

\((1 + cos A)(cos B + cos C) : (1 + cos B)(cos C + cos A) : (1 + cos C)(cos A + cos B)\)

\(a(ar - S) : b(br - S): c(cr - S)\)

\(csc B + csc C : :\)

Barycentrics

\(a2(b + c) : b2(c + a) : c2(a + b)\)

Notes

If you have The Geometer’s Sketchpad, you can view X(42).

Let A’B’C’ be the extangents triangle. Let La be the trilinear polar of A’, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, B″ = Lc∩La, C″ = La∩Lb. The lines AA″, BB″, CC″ concur in X(42). (Randy Hutson, December 26, 2015)

Let A’B’C’ be the extangents triangle. Let A″ be the trilinear product B’*C’, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(42).(Randy Hutson, December 26, 2015)

Let Ab, Ac, Bc, Ba, Ca, Cb be as defined at X(3588). Let A* be the intersection of the tangents to the Myakishev conic at Ba and Ca, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(42).(Randy Hutson, December 26, 2015)

X(42) lies on these lines: 1,2 3,967 6,31 9,941 25,41 33,393 35,58 37,210 38,518 40,581 48,197 57,1001 65,73 81,100 101,111 165,991 172,199 181,228 244,354 308,313 321,740 517,1064 560,584 649,788 694,893 748,1001 750,940 894,1045 942,1066

X(42) is the {X(1),:ref:X(43) <X(43)>}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(42), click Tables at the top of this page.

X(42) = reflection of X(321) in X(1215)

X(42) = isogonal conjugate of X(86)

X(42) = isotomic conjugate of X(310)

X(42) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,37), (6,213), (10,71), (55,228)

X(42) = crosspoint of X(i) and X(j) for these (i,j): (1,6), (33,55), (37,65)

X(42) = crosssum of X(i) and X(j) for these (i,j): (1,2), (7,77), (21,81)

X(42) = crossdifference of every pair of points on line X(239)X(514)

X(42) = circumcircle-inverse of X(32759)

X(42) = X(1)-line conjugate of X(239)

X(42) = X(i)-beth conjugate of X(j) for these (i,j): (21,551), (55,42), (100,226), (210,210), (643,171)

X(42) = bicentric sum of PU(8)

X(42) = PU(8)-harmonic conjugate of X(649)