X(53) = SYMMEDIAN POINT OF ORTHIC TRIANGLE¶

Trilinears

\(tan A cos(B - C) : tan B cos(C - A) : tan C cos(A - B)\)

Barycentrics

\(a tan A cos(B - C) : b tan B cos(C - A) : c tan C cos(A - B)\)

\((a^2 (b^2 + c^2) - (b^2 - c^2)^2)/(a^2 - b^2 - c^2) : : Let A'B'C' be the Euler triangle. Let LA be the trilinear polar of A', and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in :ref:`X(53) <X(53)>\). (Randy Hutson, June 7, 2019) X(53) lies on these lines: 4,6 5,216 25,157 30,577 45,281 115,133 128,139 137,138 141,264 232,427 273,1086 275,288 311,324 317,524 318,594 395,472 396,473 X(53) is the {X(4),:ref:X(393) <X(393)>}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(53), click Tables at the top of this page. X(53) = isogonal conjugate of X(97) <X(97)>`

Notes

Let A’B’C’ be the Euler triangle. Let LA be the trilinear polar of A’, and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(53). (Randy Hutson, June 7, 2019)

X(53) lies on these lines: 4,6 5,216 25,157 30,577 45,281 115,133 128,139 137,138 141,264 232,427 273,1086 275,288 311,324 317,524 318,594 395,472 396,473

X(53) is the {X(4),:ref:X(393) <X(393)>}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(53), click Tables at the top of this page.

X(53) = isogonal conjugate of X(97)

X(53) = isotomic conjugate of X(34386)

X(53) = anticomplement of X(34828)

X(53) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,51), (324,5)

X(53) = X(51)-cross conjugate of X(5)

X(53) = crosssum of X(3) and X(577)

X(53) = Kosnita(X(4),:ref:X(6) <X(6)>) point

X(53) = trilinear pole of line X(12077)X(15451) (the polar of X(95) wrt polar circle)

X(53) = pole wrt polar circle of trilinear polar of X(95) (line X(323)X(401))

X(53) = polar conjugate of X(95)

X(53) = excentral-to-ABC functional image of X(6)