X₉₈

TARRY POINT

X₉₈ is the {X₂,X₁₄₇}-harmonic conjugate of X₁₁₄. For a list of harmonic conjugates, click Tables at the top of this page.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₉₈ (TARRY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Central Lines: {3, 99}

• Related Centers: X₃ (CIRCUMCENTER), X₉₉ (STEINER POINT)

Trilinears: \(\sec(A + \omega ) : \sec(B + \omega ) : \sec(C + \omega )\)

Barycentrics: \(1/(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2}) : :\)

Tripolars: \(|\cos(A + \omega )| : :\)

Search Value (6, 9, 13): \(10.125996334745\)

▸ Alternate Coordinates2 forms

• Trilinears: \(bc/(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2}) : :\)

• Tripolars: \(|a (b^{4} + c^{4} - a^{2} b^{2} - a^{2} c^{2})| : :\)

▸ Linear Combinations1 combination

• X₉₈ = X₉₈ = \(2(r2 + 4rR - s2)(r2 + 2rR + s2)\) X₁ + \(3(r4 + 4Rr3 + 2r2s2 - 4rRs2 + s4)\) X₂ - \(2(r2 + 4rR - s2)(3r2 + 4rR - s2)\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines613 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, 12181}, {403, 10423}, {446, 18858}, {459, 1301}, {468, 685}, {485, 6811}, {486, 6813}, {493, 12186}, {494, 12187}, {512, 2698}, {513, 2699}, {514, 2700}, {515, 2701}, {516, 2702}, {517, 2703}, {518, 2704}, {519, 2705}, {520, 2706}, {521, 2707}, {522, 2708}, {523, 842}, {524, 2709}, {525, 2710}, {526, 9161}, {530, 9203}, {531, 9202}, {538, 9888}, {541, 9144}, {546, 18502}, {548, 38731}, {549, 8724}, {550, 38730}, {573, 3029}, {574, 1569}, {575, 3329}, {576, 7766}, {618, 21157}, {619, 21156}, {620, 631}, {635, 32552}, {636, 32553}, {637, 1306}, {638, 1307}, {648, 2452}, {675, 7432}, {689, 1799}, {693, 2856}, {729, 2422}, {732, 36517}, {733, 11620}, {739, 35353}, {741, 4444}, {759, 7427}, {783, 3505}, {801, 1368}, {804, 878}, {813, 3404}, {841, 36164}, {843, 1499}, {850, 2857}, {858, 2986}, {868, 34174}, {895, 2407}, {901, 4080}, {919, 13576}, {927, 5991}, {929, 1324}, {930, 6636}, {931, 19544}, {932, 22381}, {934, 1410}, {946, 12194}, {1029, 37456}, {1113, 2593}, {1114, 2592}, {1125, 21636}, {1131, 7374}, {1132, 7000}, {1147, 3044}, {1181, 6570}, {1196, 37892}, {1284, 19637}, {1287, 2070}, {1288, 21213}, {1290, 37959}, {1291, 5189}, {1292, 2795}, {1293, 2796}, {1294, 2797}, {1295, 2798}, {1297, 2799}, {1302, 1995}, {1309, 17987}, {1311, 7439}, {1316, 5191}, {1342, 5404}, {1343, 5403}, {1350, 5969}, {1351, 14614}, {1370, 6504}, {1379, 3414}, {1380, 3413}, {1385, 29299}, {1423, 7350}, {1447, 7061}, {1478, 10802}, {1479, 10801}, {1482, 29269}, {1498, 13380}, {1576, 7668}, {1587, 18993}, {1588, 18994}, {1593, 5186}, {1598, 10790}, {1656, 7942}, {1657, 38733}, {1687, 2009}, {1688, 2010}, {1698, 29117}, {1699, 10789}, {1756, 7095}, {1764, 38481}, {1989, 23969}, {2051, 4279}, {2086, 8789}, {2222, 5143}, {2319, 6210}, {2378, 5917}, {2379, 5916}, {2387, 11674}, {2482, 3524}, {2558, 6177}, {2559, 6178}, {2691, 36001}, {2696, 7464}, {2711, 3309}, {2712, 3667}, {2713, 6000}, {2714, 6001}, {2720, 5061}, {2770, 4108}, {2777, 16278}, {2855, 3260}, {2858, 3266}, {3068, 8980}, {3069, 13967}, {3070, 14245}, {3071, 14231}, {3090, 6722}, {3091, 5395}, {3095, 7760}, {3098, 6194}, {3146, 38259}, {3153, 11635}, {3222, 3504}, {3225, 6234}, {3229, 8841}, {3269, 18338}, {3291, 9091}, {3314, 34507}, {3316, 10783}, {3317, 10784}, {3406, 7746}, {3407, 3818}, {3424, 9752}, {3428, 22514}, {3430, 8669}, {3456, 3518}, {3507, 23605}, {3510, 8923}, {3512, 7351}, {3522, 10992}, {3523, 38748}, {3525, 20399}, {3526, 7930}, {3533, 38746}, {3534, 33638}, {3543, 9880}, {3545, 5461}, {3563, 16230}, {3566, 23700}, {3574, 12208}, {3576, 11711}, {3579, 29151}, {3597, 5706}, {3616, 11724}, {3655, 9884}, {3734, 35925}, {3757, 38470}, {3815, 5038}, {3830, 17503}, {3832, 18845}, {3843, 18501}, {3849, 13241}, {3851, 15092}, {3855, 18843}, {3934, 5149}, {3972, 35930}, {4232, 9064}, {4239, 9058}, {4297, 28483}, {4558, 25051}, {4588, 30588}, {4995, 12350}, {5000, 34240}, {5001, 34239}, {5008, 22521}, {5020, 37874}, {5025, 9863}, {5026, 5085}, {5033, 7612}, {5034, 5477}, {5039, 5304}, {5050, 11174}, {5055, 22566}, {5071, 14971}, {5092, 8289}, {5169, 7578}, {5188, 6308}, {5210, 8719}, {5217, 15452}, {5254, 34870}, {5286, 13356}, {5293, 29052}, {5298, 12351}, {5306, 5480}, {5465, 10706}, {5469, 5479}, {5470, 5478}, {5475, 11170}, {5487, 22532}, {5488, 22531}, {5569, 19911}, {5587, 10791}, {5597, 12179}, {5598, 12180}, {5603, 10800}, {5657, 29127}, {5663, 9160}, {5840, 10769}, {5870, 14229}, {5871, 14244}, {5886, 29111}, {5913, 35188}, {5914, 9136}, {5965, 7779}, {5968, 34810}, {5970, 32472}, {5971, 6082}, {5978, 5982}, {5979, 5983}, {5996, 32730}, {6002, 12031}, {6011, 30273}, {6072, 22103}, {6199, 10839}, {6200, 10848}, {6201, 10793}, {6202, 10792}, {6221, 10841}, {6222, 8304}, {6233, 7610}, {6245, 12196}, {6246, 12198}, {6247, 12202}, {6249, 12206}, {6250, 12211}, {6251, 12210}, {6284, 13183}, {6295, 14539}, {6312, 6320}, {6316, 6319}, {6395, 10840}, {6396, 10847}, {6398, 10842}, {6399, 8305}, {6433, 10843}, {6434, 10844}, {6435, 10849}, {6436, 10850}, {6530, 16318}, {6539, 8701}, {6564, 35767}, {6565, 35766}, {6578, 32014}, {6582, 14538}, {6625, 7379}, {6655, 32152}, {6670, 36765}, {6684, 29119}, {6784, 6785}, {6795, 9832}, {6995, 8796}, {7166, 8926}, {7168, 8927}, {7354, 13182}, {7391, 13579}, {7414, 26706}, {7426, 9060}, {7438, 9107}, {7448, 9083}, {7449, 9056}, {7453, 9057}, {7458, 9061}, {7472, 38702}, {7494, 26870}, {7505, 26269}, {7607, 37637}, {7697, 10000}, {7751, 8178}, {7778, 15069}, {7781, 18768}, {7788, 11898}, {7790, 37242}, {7797, 37336}, {7818, 34623}, {7824, 13334}, {7827, 37345}, {7857, 37466}, {7875, 38317}, {7887, 10349}, {7901, 10333}, {7907, 10131}, {7911, 32151}, {7919, 9996}, {7932, 10345}, {8059, 8808}, {8150, 12252}, {8196, 11837}, {8203, 11838}, {8212, 11840}, {8213, 11841}, {8229, 24624}, {8295, 9865}, {8296, 10853}, {8297, 10854}, {8375, 10837}, {8376, 10838}, {8587, 8590}, {8591, 10304}, {8600, 11184}, {8690, 16434}, {8693, 37675}, {8704, 9831}, {8770, 34808}, {8784, 11606}, {8859, 11645}, {8997, 9540}, {9059, 26266}, {9066, 26233}, {9069, 26235}, {9070, 26227}, {9079, 26278}, {9080, 26276}, {9087, 9465}, {9123, 9215}, {9150, 34087}, {9167, 15702}, {9184, 9191}, {9221, 15032}, {9300, 12007}, {9307, 9308}, {9381, 21284}, {9413, 9414}, {9479, 14718}, {9738, 33371}, {9739, 33370}, {9746, 28841}, {9875, 34628}, {9927, 12193}, {9955, 29293}, {9956, 29155}, {10098, 10295}, {10101, 37979}, {10102, 36168}, {10113, 12201}, {10166, 31839}, {10264, 15545}, {10290, 32515}, {10299, 35022}, {10303, 38751}, {10310, 13173}, {10313, 35098}, {10486, 17005}, {10519, 15589}, {10531, 10803}, {10532, 10804}, {10733, 12066}, {10794, 10893}, {10795, 10894}, {10797, 10895}, {10798, 10896}, {10810, 22564}, {10997, 35375}, {11121, 36515}, {11122, 36514}, {11152, 11155}, {11172, 23055}, {11231, 29129}, {11248, 13189}, {11249, 13190}, {11414, 13175}, {11490, 11496}, {11538, 37349}, {11602, 23004}, {11603, 23005}, {11669, 31489}, {11822, 13176}, {11823, 13177}, {11826, 13180}, {11827, 13181}, {11828, 13184}, {11829, 13185}, {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}

▸ Functional Relations67 relations

• X₉₈ = midpoint between X₂₀ and X₁₄₈

• X₉₈ = reflection of X_i in X_j for these (i,j): (4, 115), (99, 3), (147, 114), (1513, 230)

• X₉₈ = isogonal conjugate of X₅₁₁

• X₉₈ = isotomic conjugate of X₃₂₅

• X₉₈ = complement of X₁₄₇

• X₉₈ = anticomplement of X₁₁₄

• X₉₈ = circumcircle antipode of X₉₉

• X₉₈ = X₂₉₀-Ceva conjugate of X₂₈₇

• X₉₈ = cevapoint of X_i and X_j for these (i,j): (2, 385), (6, 237)

• X₉₈ = X_i-cross conjugate of X_j for these (i,j): (230, 2), (237, 6), (248, 287), (446, 511)

• X₉₈ = crosssum of X₃₈₅ and X₄₀₁

• X₉₈ = X₂-Hirst inverse of X₂₈₇

• X₉₈ = perspector of ABC and triangle formed by Lemoine axis (or PU(1) or PU(2)) reflected in sides of ABC

• X₉₈ = Λ(X₄, X₆₉) (the line that is the isotomic conjugate of the Jerabek hyperbola)

• X₉₈ = trilinear pole of line X₆X₅₂₃ (polar of X₂₉₇ wrt polar circle, and the radical axis of circles with segments X₁₃X₁₆ and X₁₄X₁₅ as diameters)

• X₉₈ = pole wrt polar circle of trilinear polar of X₂₉₇ (line X₁₁₄X₁₃₂)

• X₉₈ = pole wrt {circumcircle, nine-point circle}-inverter of line X₁₁₅X₁₂₅

• X₉₈ = X₄₈-isoconjugate (polar conjugate) of X₂₉₇

• X₉₈ = X₆-isoconjugate of X₁₉₅₉

• X₉₈ = inverse-in-polar-circle of X₁₃₂

• X₉₈ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₂₅

• X₉₈ = inverse-in-circle-{X₁₆₈₇,X₁₆₈₈,PU(1),PU(2)}} of X₂₇₁₅

• X₉₈ = Ψ(X₆, X₅₂₃)

• X₉₈ = Ψ(X₁₉₀, X₇₁)

• X₉₈ = Kiepert-hyperbola antipode of X₄

• X₉₈ = reflection of X₈₄₂ in the Euler line

• X₉₈ = reflection of X₂₆₉₈ in the Brocard axis

• X₉₈ = reflection of X₂₆₉₉ in line X₁X₃

• X₉₈ = X₁₂₉-of-excentral-triangle

• X₉₈ = X₁₃₀-of-hexyl-triangle

• X₉₈ = X₃-of-1st-anti-Brocard-triangle

• X₉₈ = perspector of ABC and 1st Neuberg triangle

• X₉₈ = trilinear product of vertices of 1st Neuberg triangle

• X₉₈ = orthocenter of X₁₃X₁₄X₂₃₉₄

• X₉₈ = 2nd-Parry-to-ABC similarity image of X₂

• X₉₈ = trilinear product of PU(88)

• X₉₈ = X₂₄₅₆ of 6th Brocard triangle

• X₉₈ = midpoint of PU(135)

• X₉₈ = bicentric sum of PU(135)

• X₉₈ = perspector of ABC and circumsymmedial triangle of Artzt triangle

• X₉₈ = McCay-to-Artzt similarity image of X₃₈₁

• X₉₈ = circumcircle-antipode of X₉₉

• X₉₈ = the point of intersection, other than A, B, and C, of the circumcircle and Kiepert hyperbola

• X₉₈ = Ψ(X₁₀₁, X₁₀₀

• X₉₈ = Λ(X₃, X₆)

• X₉₈ = homothetic center of 5th anti-Brocard triangle and Euler triangle

• X₉₈ = Thomson-isogonal conjugate of X₅₁₂

• X₉₈ = Lucas-isogonal conjugate of X₅₁₂

• X₉₈ = X₁₃₈₀-of-circummedial-triangle

• X₉₈ = X₆₂₃₃-of-circumsymmedial-triangle

• X₉₈ = Cundy-Parry Phi transform of X₇₆

• X₉₈ = Cundy-Parry Psi transform of X₃₂

• X₉₈ = perspector of ABC and vertex-triangle of reflection triangles of PU(1)

• X₉₈ = X₂-of-2nd-anti-Parry-triangle

• X₉₈ = excentral-to-ABC functional image of X₁₂₈₂

• X₉₈ = orthic-to-ABC functional image of X₁₂₉

• X₉₈ = Ψ(X₁, X₈₁₀)

• X₉₈ = Ψ(X₈₁₁, X₁)

• X₉₈ = barycentric product of circumcircle intercepts of line X₂X₆₄₇

• X₉₈ = trilinear pole, wrt Thomson triangle, of line X₂X₁₃₅₀

• X₉₈ = areal center of pedal triangles of PU(1)

• X₉₈ = orthocenter of X₉₈X₉₉X₃₁₉₅₃

• X₉₈ = orthocenter of X₁₁₄X₁₁₅X₃₁₉₅₃

• X₉₈ = X₂-Ceva conjugate of X₃₆₈₉₉

• X₉₈ = perspector of circumconic centered at X₃₆₈₉₉

• X₉₈ = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A, B, C,X₃,X₂₅,X₃₂}} (isogonal conjugate of line X₄X₆₉)

• X₉₈ = BSS(a→a^2) of X₆₇₃

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 H_a be the orthocenter of BA'C, and define H_b and H_c cyclically. The lines AH_a, BH_b, CH_c concur in X₉₈. (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 O_a be the circumcenter of BA'C, and define O_b and O_c cyclically. The lines AO_a, BO_b, CO_c concur in X₉₈. (Randy Hutson, July 20, 2016)

J. W. 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. X₉₈ is the radical center of the circumcircles of AA'A″, BB'B″, CC'C″. (Randy Hutson, November 18, 2015)

Let H_aH_bH_c be the orthic triangle. Let L_a be the Lemoine axis of triangle AH_bH_c, and define L_b and L_c cyclically. Let A″ = L_b∩L_c, and define B″ and C″ cyclically. Triangle A″B″C″ is inversely similar to ABC, with similicenter X₂. The lines AA″, BB″, CC″ concur in X₉₈, which is X₄-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 X₂. When P = X₆₇₁, Q = X₉₈. See also X₆₀₅₄ and X₂₄₈₀₈. (Randy Hutson, October 15, 2018)