X(98) = TARRY POINT¶
Trilinears
\(sec(A + ω) : sec(B + ω) : sec(C + ω)\)
\(bc/(b4 + c4 - a2b2 - a2c2) : :\)
Barycentrics
\(1/(b4 + c4 - a2b2 - a2c2) : :\)
Notes
Let A’ be the apex of the isosceles triangle BA’C constructed inward on BC such that ∠A’BC = ∠A’CB = ω. Define B’ nad C’ cyclically. Let Ha be the orthocenter of BA’C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(98). (Randy Hutson, July 20, 2016)
Let A’ be the apex of the isosceles triangle BA’C constructed outward on BC such that ∠A’BC = ∠A’CB = ω/2. Define B’ and C’ cyclically. Let Oa be the circumcenter of BA’C, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(98). (Randy Hutson, July 20, 2016)
If you have The Geometer’s Sketchpad, you can view X(98).
Clawson, “Points on the circumcircle,” American Mathematical Monthly 32 (1925) 169-174.
Let A’B’C’ be the 1st Brocard triangle. Let A″ be the reflection of A in B’C’, and define B″ and C″ cyclically. X(98) is the radical center of the circumcircles of AA’A″, BB’B″, CC’C″. (Randy Hutson, November 18, 2015)
Let A’B’C’ be the orthic triangle. Let La be the Lemoine axis of triangle AB’C’, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. Triangle A″B″C″ is inversely similar to ABC, with similicenter X(2). The lines AA″, BB″, CC″ concur in X(98), which is X(4)-of-A″B″C″. (Randy Hutson, July 31 2018)
Let P be a point on the Steiner circumellipse. Let A’ be the orthocenter of BCP, and define B’ and C’ cyclically. Let Q be the centroid of A’B’C’. The locus of Q as P varies is an ellipse similar and orthogonal to the Steiner circumellipse, and also centered at X(2). When P = X(671), Q = X(98). See also X(6054) and X(24808). (Randy Hutson, October 15, 2018)
X(98) lies on these lines: {1, 7970}, {2, 110}, {3, 76}, {4, 32}, {5, 83}, {6, 262}, {10, 101}, {11, 10768}, {12, 10799}, {13, 1080}, {14, 383}, {15, 33388}, {16, 33389}, {17, 6115}, {18, 6114}, {20, 148}, {22, 925}, {23, 94}, {24, 1289}, {25, 107}, {26, 1286}, {30, 671}, {35, 10086}, {36, 10089}, {39, 3399}, {40, 6010}, {54, 3203}, {55, 3027}, {56, 3023}, {57, 24469}, {61, 22691}, {62, 22692}, {69, 13355}, {74, 690}, {95, 20775}, {96, 32692}, {100, 228}, {102, 2785}, {103, 2786}, {104, 2787}, {105, 2788}, {106, 2789}, {108, 1402}, {109, 171}, {111, 1637}, {113, 13193}, {119, 13194}, {127, 12207}, {140, 7832}, {141, 19120}, {165, 13174}, {185, 1303}, {186, 935}, {187, 11676}, {194, 9737}, {230, 1503}, {235, 11380}, {237, 6037}, {250, 34978}, {251, 30505}, {264, 3425}, {265, 14849}, {275, 427}, {316, 15980}, {325, 2065}, {336, 1310}, {338, 1632}, {355, 12195}, {371, 13885}, {372, 13938}, {376, 543}, {378, 30247}, {381, 598}, {382, 22515}, {384, 6248}, {385, 511}, {386, 34454}, {402, 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{11839, 11897}, {12092, 31074}, {12112, 32732}, {12151, 22110}, {12200, 12599}, {12209, 12600}, {12245, 28532}, {12258, 31162}, {12355, 15681}, {12699, 29091}, {12702, 29177}, {13449, 14041}, {13580, 14808}, {13581, 14807}, {13672, 13687}, {13748, 14234}, {13749, 14238}, {13792, 13807}, {13935, 13989}, {14060, 14570}, {14236, 23261}, {14240, 23251}, {14458, 36990}, {14485, 18907}, {14534, 37360}, {14540, 22914}, {14541, 22869}, {14561, 16989}, {14575, 36794}, {14630, 14633}, {14631, 14632}, {14644, 15359}, {14719, 34783}, {14850, 38728}, {15054, 31854}, {15182, 19379}, {15246, 20189}, {15300, 19708}, {15357, 18331}, {15407, 34138}, {15483, 21163}, {15526, 34841}, {15598, 21167}, {15682, 32532}, {15687, 33698}, {15694, 26614}, {15698, 36521}, {15709, 22247}, {16083, 16089}, {16085, 35574}, {16086, 29241}, {16115, 16125}, {16166, 18366}, {16167, 37980}, {16188, 36173}, {16984, 18553}, {17008, 37182}, {17734, 32682}, {17758, 21554}, {17985, 36067}, {18440, 37071}, {18446, 36516}, {18806, 35385}, {18906, 35387}, {18931, 36893}, {19179, 31804}, {19642, 37443}, {19649, 34594}, {20126, 36826}, {20187, 21312}, {20190, 36811}, {20404, 32305}, {22143, 36841}, {22239, 37777}, {22467, 26179}, {22498, 35422}, {22520, 22753}, {22522, 22831}, {22523, 22832}, {22524, 22833}, {22791, 29265}, {23098, 36214}, {24815, 24828}, {24975, 25328}, {25213, 38403}, {25216, 38404}, {25406, 34229}, {25446, 32022}, {26273, 28476}, {26326, 26379}, {26327, 26403}, {26328, 26427}, {26329, 26428}, {26330, 26429}, {26331, 26430}, {26332, 26431}, {26333, 26432}, {26446, 29030}, {26613, 37461}, {27797, 28210}, {28474, 35103}, {28475, 35107}, {29067, 30115}, {30241, 37620}, {30670, 31394}, {30716, 38552}, {32111, 32681}, {32242, 32274}, {32335, 32369}, {32522, 33004}, {33512, 38793}, {33640, 37453}, {34133, 34134}, {34310, 35189}, {34396, 37988}, {34475, 37508}, {35705, 37451}, {36036, 36066}, {36071, 36104}, {36495, 36511}, {36526, 36548}, {36557, 36580}, {38253, 38282}
X(98) is the {X(2),:ref:X(147) <X(147)>}-harmonic conjugate of X(114). For a list of harmonic conjugates, click Tables at the top of this page.
X(98) = midpoint between X(20) and X(148)
X(98) = reflection of X(i) in X(j) for these (i,j): (4,115), (99,3), (147,114), (1513,230)
X(98) = isogonal conjugate of X(511)
X(98) = isotomic conjugate of X(325)
X(98) = complement of X(147)
X(98) = anticomplement of X(114)
X(98) = circumcircle antipode of X(99)
X(98) = X(290)-Ceva conjugate of X(287)
X(98) = cevapoint of X(i) and X(j) for these (i,j): (2,385), (6,237)
X(98) = X(i)-cross conjugate of X(j) for these (i,j): (230,2), (237,6), (248,287), (446,511)
X(98) = crosssum of X(385) and X(401)
X(98) = X(2)-Hirst inverse of X(287)
X(98) = perspector of ABC and triangle formed by Lemoine axis (or PU(1) or PU(2)) reflected in sides of ABC
X(98) = Λ(X(4), X(69)) (the line that is the isotomic conjugate of the Jerabek hyperbola)
X(98) = trilinear pole of line X(6)X(523) (polar of X(297) wrt polar circle, and the radical axis of circles with segments X(13)X(16) and X(14)X(15) as diameters)
X(98) = pole wrt polar circle of trilinear polar of X(297) (line X(114)X(132))
X(98) = pole wrt {circumcircle, nine-point circle}-inverter of line X(115)X(125)
X(98) = X(48)-isoconjugate (polar conjugate) of X(297)
X(98) = X(6)-isoconjugate of X(1959)
X(98) = inverse-in-polar-circle of X(132)
X(98) = inverse-in-{circumcircle, nine-point circle}-inverter of X(125)
X(98) = inverse-in-circle-{X(1687),:ref:X(1688) <X(1688)>,PU(1),PU(2)}} of X(2715)
X(98) = Kiepert-hyperbola antipode of X(4)
X(98) = reflection of X(842) in the Euler line
X(98) = reflection of X(2698) in the Brocard axis
X(98) = reflection of X(2699) in line X(1)X(3)
X(98) = X(129)-of-excentral-triangle
X(98) = X(130)-of-hexyl-triangle
X(98) = X(3)-of-1st-anti-Brocard-triangle
X(98) = perspector of ABC and 1st Neuberg triangle
X(98) = trilinear product of vertices of 1st Neuberg triangle
X(98) = orthocenter of X(13)X(14)X(2394)
X(98) = 2nd-Parry-to-ABC similarity image of X(2)
X(98) = trilinear product of PU(88)
X(98) = X(2456) of 6th Brocard triangle
X(98) = midpoint of PU(135)
X(98) = bicentric sum of PU(135)
X(98) = perspector of ABC and circumsymmedial triangle of Artzt triangle
X(98) = McCay-to-Artzt similarity image of X(381)
X(98) = circumcircle-antipode of X(99)
X(98) = the point of intersection, other than A, B, and C, of the circumcircle and Kiepert hyperbola
X(98) = homothetic center of 5th anti-Brocard triangle and Euler triangle
X(98) = Thomson-isogonal conjugate of X(512)
X(98) = Lucas-isogonal conjugate of X(512)
X(98) = X(1380)-of-circummedial-triangle
X(98) = X(6233)-of-circumsymmedial-triangle
X(98) = Cundy-Parry Phi transform of X(76)
X(98) = Cundy-Parry Psi transform of X(32)
X(98) = perspector of ABC and vertex-triangle of reflection triangles of PU(1)
X(98) = X(2)-of-2nd-anti-Parry-triangle
X(98) = excentral-to-ABC functional image of X(1282)
X(98) = orthic-to-ABC functional image of X(129)
X(98) = barycentric product of circumcircle intercepts of line X(2)X(647)
X(98) = trilinear pole, wrt Thomson triangle, of line X(2)X(1350)
X(98) = areal center of pedal triangles of PU(1)
X(98) = orthocenter of X(98)X(99)X(31953)
X(98) = orthocenter of X(114)X(115)X(31953)
X(98) = X(2)-Ceva conjugate of X(36899)
X(98) = perspector of circumconic centered at X(36899)
X(98) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,:ref:X(3) <X(3)>,:ref:X(25) <X(25)>,:ref:X(32) <X(32)>}} (isogonal conjugate of line X(4)X(69))
X(98) = BSS(a→a^2) of X(673)