X(58) = X(58) ISOGONAL CONJUGATE OF X(10)¶
Trilinears
\(a/(b + c) : b/(c + a) : c/(a + b)\)
\((1 - cos A)/(cos B + cos C) : :\)
\(sa2 + SR : sb2 + SR : sc2 + SR\)
\(r cos A - s sin A : : , where s = semiperimeter and r = inradius\)
\(sin(A - U) : : , U as at :ref:`X(572) <X(572)>\) and X(573) <X(573)>`
\((R/r) - 1/(cos B + cos C) : :\)
\((r/R) - cos 2A + 1 : :\)
\(eccentricity of A-Soddy ellipse : :\)
Barycentrics
\(a2/(b + c) : b2/(c + a) : c2/(a + b) :ref:`X(58) <X(58)>\) is the point of concurrence of the Brocard axes of triangles BIC, CIA, AIB, ABC, (where I denotes the incenter, X(1)), as proved in Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines, Forum Geometricorum 1 (2001) 59-68. Let (Sa) be the reflection of the Spieker circle in BC, and define (Sb), (Sc) cyclically. X(58) is the radical center of (Sa), (Sb), (Sc). (Randy Hutson, July 20, 2016) Let A’B’C’ be the anticomplement of the Feuerbach triangle. Let A″B″C″ be the circumcevian triangle, wrt A’B’C’, of X(1). The lines AA″, BB″, CC″ concur in X(58). (Randy Hutson, July 20, 2016) Let A’B’C’ be the Feuerbach triangle. Let La be the tangent to the nine-point circle at A’, and define Lb, Lc cyclically. Let A″ be the isogonal conjugate of the trilinear pole of La, and define B″, C″ cyclically. Let A* = BB″∩CC″, B* = CC″∩AA″, C* = AA″∩BB″. The lines AA*, BB*, CC* concur in X(58). (Randy Hutson, July 20, 2016) Let A’B’C’ be the 2nd circumperp triangle. Let A″ be the trilinear product B’*C’, and define B″, C″ cyclically. A″, B″, C″ are collinear on line X(36)X(238) (the trilinear polar of X(81)). The lines AA″, BB″, CC″ concur in X(58). (Randy Hutson, July 20, 2016) For an artistic design generated by X(58), see X(244). X(58) lies on these lines: 1,21 2,540 3,6 7,272 8,996 9,975 10,171 20,387 25,967 27,270 28,34 29,162 35,42 36,60 40,601 41,609 43,979 46,998 56,222 65,109 82,596 84,990 86,238 87,978 99,727 101,172 103,112 106,110 229,244 269,1014 274,870 314,987 405,940 519,1043 942,1104 977,982 1019,1027 X(58) is the {X(3),:ref:X(6) <X(6)>}-harmonic conjugate of X(386). For a list of other harmonic conjugates of X(58), click Tables at the top of this page. X(58) = isogonal conjugate of X(10) <X(10)>`
Notes
X(58) is the point of concurrence of the Brocard axes of triangles BIC, CIA, AIB, ABC, (where I denotes the incenter, X(1)), as proved in Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines, Forum Geometricorum 1 (2001) 59-68.
Let (Sa) be the reflection of the Spieker circle in BC, and define (Sb), (Sc) cyclically. X(58) is the radical center of (Sa), (Sb), (Sc). (Randy Hutson, July 20, 2016)
Let A’B’C’ be the anticomplement of the Feuerbach triangle. Let A″B″C″ be the circumcevian triangle, wrt A’B’C’, of X(1). The lines AA″, BB″, CC″ concur in X(58). (Randy Hutson, July 20, 2016)
Let A’B’C’ be the Feuerbach triangle. Let La be the tangent to the nine-point circle at A’, and define Lb, Lc cyclically. Let A″ be the isogonal conjugate of the trilinear pole of La, and define B″, C″ cyclically. Let A* = BB″∩CC″, B* = CC″∩AA″, C* = AA″∩BB″. The lines AA*, BB*, CC* concur in X(58). (Randy Hutson, July 20, 2016)
Let A’B’C’ be the 2nd circumperp triangle. Let A″ be the trilinear product B’*C’, and define B″, C″ cyclically. A″, B″, C″ are collinear on line X(36)X(238) (the trilinear polar of X(81)). The lines AA″, BB″, CC″ concur in X(58). (Randy Hutson, July 20, 2016)
For an artistic design generated by X(58), see X(244).
X(58) lies on these lines: 1,21 2,540 3,6 7,272 8,996 9,975 10,171 20,387 25,967 27,270 28,34 29,162 35,42 36,60 40,601 41,609 43,979 46,998 56,222 65,109 82,596 84,990 86,238 87,978 99,727 101,172 103,112 106,110 229,244 269,1014 274,870 314,987 405,940 519,1043 942,1104 977,982 1019,1027
X(58) is the {X(3),:ref:X(6) <X(6)>}-harmonic conjugate of X(386). For a list of other harmonic conjugates of X(58), click Tables at the top of this page.
X(58) = isogonal conjugate of X(10)
X(58) = isotomic conjugate of X(313)
X(58) = complement of X(1330)
X(58) = anticomplement of X(3454)
X(58) = circumcircle-inverse of X(1326)
X(58) = Brocard-circle-inverse of X(386)
X(58) = X(i)-Ceva conjugate of X(j) for these (i,j): (81,284), (267,501), (270,28)
X(58) = cevapoint of X(6) and X(31)
X(58) = X(i)-cross conjugate of X(j) for these (i,j): (6,81), (36,106), (56,28), (513,109)
X(58) = crosspoint of X(i) and X(j) for these (i,j): (1,267), (21,285), (27,86), (60,270)
X(58) = crosssum of X(i) and X(j) for these (i,j): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227), (594, 756)
X(58) = crossdifference of every pair of points on line X(523)X(661)
X(58) = X(6)-Hirst inverse of X(1326)
X(58) = antigonal conjugate of isogonal conjugate of X(1324)
X(58) = antigonal conjugate of isotomic conjugate of X(21277)
X(58) = antigonal conjugate of polar conjugate of X(37770)
X(58) = X(i)-beth conjugate of X(j) for these (i,j): (21,21), (60,58), (110,58), (162,58), (643,58), (1098,283)
X(58) = barycentric product of PU(31)
X(58) = trilinear pole of line X(649)X(834)
X(58) = {X(1),:ref:X(31) <X(31)>}-harmonic conjugate of X(595)
X(58) = {X(21),:ref:X(283) <X(283)>}-harmonic conjugate of X(2328)
X(58) = X(42)-isoconjugate of X(75)
X(58) = X(71)-isoconjugate of X(92)
X(58) = X(101)-isoconjugate of X(1577)
X(58) = homothetic center of 2nd circumperp triangle and ‘Hatzipolakis-Brocard triangle’ (A’B’C’ as defined at X(5429))
X(58) = trilinear product of vertices of 2nd circumperp triangle
X(58) = perspector of 2nd circumperp triangle and unary cofactor triangle of 1st circumperp triangle
X(58) = perspector of ABC and cross-triangle of ABC and 2nd circumperp triangle
X(58) = Cundy-Parry Phi transform of X(573)
X(58) = Cundy-Parry Psi transform of X(13478)
X(58) = perspector of ABC and unary cofactor triangle of Gemini triangle 11
X(58) = {X(1),:ref:X(21) <X(21)>}-harmonic conjugate of X(4653)