X(51) = CENTROID OF ORTHIC TRIANGLE¶

Trilinears

\(a2cos(B - C) : :\)

\(a[a2(b2 + c2) - (b2 - c2)2] : :\)

\(sin A (sin 2B + sin 2C) : :\)

\(sec A (csc 2B + csc 2C) : :\)

Barycentrics

\(a3cos(B - C) : b3cos(C - A) : c3cos(A - B)\)

Notes

Let A’B’C’ be the anticomplementary triangle and let Ba and Ca be the orthogonal projections of B’ and C’ on BC, respectively. Define Cb and Ac cyclically, and define Ab and Bc cyclically. Then X(51) is the centroid of BaCaCbAbAcBc. (Randy Hutson, April 9, 2016)

Let L be the van Aubel line. Let U = X(6)X(25), the isogonal conjugate of polar conjugate of L; let V = X(4)X(51), the polar conjugate of the isogonal conjugate of L. Then

X(51) = U∩V. (Randy Hutson, April 9, 2016) Let A’B’C’ be the orthic triangle. Let Oa be the A-McCay circle of triangle AB’C’, and define Ob, Oc cyclically. X(51) is the radical center of circles Oa, Ob, Oc. (Randy Hutson, July 31 2018)

X(51) lies on these lines: 2,262 4,185 5,52 6,25 21,970 22,182 23,575 24,578 26,569 31,181 39,237 44,209 54,288 107,275 125,132 129,137 130,138 199,572 210,374 216,418 381,568 397,462 398,463 573,1011

X(51) is the {X(5),:ref:X(143) <X(143)>}-harmonic conjugate of X(52). For a list of other harmonic conjugates of X(51), click Tables at the top of this page.

X(51) = reflection of X(210) in X(375)

X(51) = isogonal conjugate of X(95)

X(51) = isotomic conjugate of X(34384)

X(51) = complement of X(2979)

X(51) = anticomplement of X(3819)

X(51) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,53), (5,216), (6,217)

X(51) = X(2)-Ceva conjugate of complementary conjugate of X(34845)

X(51) = X(217)-cross conjugate of X(216)

X(51) = crosspoint of X(i) and X(j) for these (i,j): (4,6), (5,53)

X(51) = crosssum of X(i) and X(j) for these (i,j): (2,3), (6,160), (54,97)

X(51) = crossdifference of every pair of points on line X(323)X(401)

X(51) = inverse-in-orthosymmedial-circle of X(125)