X351

CENTER OF THE PARRY CIRCLE

X351 is the center of the Parry circle introduced in TCCT (Art. 8.13) as the circle that passes through Xi for I = 2, 15, 16, 23, 110, 111, 352, 353.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 187, 237 }: Central Line L(187, 237) { 110, 526 }: Central Line L(110, 526) { 2, 804 }: Central Line L(2, 804) { 184, 686 }: Central Line L(184, 686) { 694, 881 }: Central Line L(694, 881) { 865, 888 }: Central Line L(865, 888) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_187 = INVERSE-IN-CIRCUMCIRCLE OF X(6) (SCHOUTE CENTER) (1.1986, 1.2559) X_237 = -LINE CONJUGATE OF X(2) (0.9894, 1.3825) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_526 = ISOGONAL CONJUGATE OF X(476) (1.0501, 0.4856) X_2 = CENTROID (0.5374, 0.2000) X_804 = ODD (- 2, 2) INFINITY POINT (0.9171, 0.6474) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_686 = CROSSDIFFERENCE OF X(4) AND X(110) (0.1521, -0.4080) X_694 = ISOGONAL CONJUGATE OF X(385) (1.0179, 0.0378) X_881 = ISOGONAL CONJUGATE OF X(880) (0.9993, -0.0247) X_865 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(512) (-0.1837, -1.6869) X_888 = ISOTOMIC CONJUGATE OF X(886) (0.2587, -0.3165) X_351 = CENTER OF THE PARRY CIRCLE (1.3537, 1.1619) A B C X187 X237 X110 X526 X2 X804 X184 X686 X694 X881 X865 X888 X351 (a, b, c) = (5, 6, 7) | AB = 1

X351 (CENTER OF THE PARRY CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{2} - c^{2})(b^{2} + c^{2} - 2a^{2})\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{2}(b^{2} - c^{2})(b^{2} + c^{2} - 2a^{2})\)
Search Value (6, 9, 13): \(-2.815203880078\)

X351 lies on the cubic K978.

▸ Incident Central Lines130 lines

{2, 804}, {3, 2780}, {6, 9023}, {15, 9162}, {16, 9163}, {23, 9213}, {25, 14998}, {30, 19912}, {37, 17989}, {42, 17990}, {51, 14397}, {110, 526}, {111, 2492}, {184, 686}, {187, 237}, {251, 17997}, {308, 17995}, {352, 9212}, {353, 13242}, {498, 31502}, {513, 9810}, {514, 23817}, {520, 13302}, {522, 49628}, {523, 4108}, {542, 36255}, {650, 50538}, {670, 23356}, {690, 1641}, {691, 34539}, {694, 881}, {740, 4763}, {812, 10180}, {850, 44451}, {865, 888}, {878, 1976}, {900, 9978}, {924, 14696}, {1304, 32696}, {1400, 17992}, {1499, 9127}, {1510, 13315}, {1635, 1962}, {1636, 23611}, {1637, 8029}, {2054, 17991}, {2395, 46316}, {2444, 14908}, {2514, 21006}, {2793, 6055}, {2799, 11123}, {2854, 40282}, {2872, 7600}, {2881, 9157}, {3268, 9479}, {3455, 14339}, {3566, 39904}, {3900, 13254}, {4455, 8034}, {4728, 53034}, {4809, 6370}, {4893, 9279}, {5020, 44817}, {5054, 16235}, {5466, 8587}, {5652, 16508}, {5663, 19902}, {5926, 9128}, {5996, 25423}, {6091, 34519}, {6131, 7665}, {6353, 47206}, {6368, 32407}, {6562, 6587}, {7664, 22105}, {8429, 35901}, {8672, 47826}, {8675, 46953}, {8704, 31772}, {8723, 33972}, {9009, 17415}, {9150, 18829}, {9158, 16171}, {9186, 9192}, {9194, 27550}, {9195, 27551}, {9200, 13305}, {9201, 13304}, {9429, 14406}, {9517, 45082}, {10166, 11171}, {10567, 20976}, {11205, 14403}, {11215, 31986}, {11616, 11620}, {13223, 13224}, {13263, 13264}, {13284, 13285}, {13290, 13291}, {13308, 13309}, {13316, 13319}, {13317, 13320}, {13718, 13719}, {13841, 13842}, {14084, 14830}, {14272, 47139}, {14277, 18310}, {14315, 48024}, {14316, 50545}, {14337, 14338}, {14698, 45147}, {16156, 16157}, {17999, 21448}, {19901, 33962}, {22984, 22985}, {23035, 23036}, {24809, 24810}, {27798, 45675}, {27811, 47776}, {28374, 29198}, {31176, 32472}, {31953, 47200}, {36213, 38366}, {39232, 46276}, {41079, 47173}, {41300, 47128}, {44427, 47217}, {44889, 46130}, {46127, 46131}, {51335, 52743}

▸ Functional Relations48 relations

  • X351 = midpoint of Xi and Xj for these {i,j}: {2, 9147}, {15, 9162}, {16, 9163}, {23, 9213}, {110, 9138}, {111, 9156}, {352, 9212}, {353, 13242}, {647, 8644}, {669, 17414}, {1635, 1962}, {3288, 11186}, {3569, 9135}, {4108, 36900}, {5027, 9208}, {5466, 9485}, {5638, 5639}, {9123, 9185}, {9126, 11615}, {9131, 9979}, {9157, 13114}, {9200, 13305}, {9201, 13304}, {9810, 9811}, {9978, 9980}, {11123, 14420}, {11616, 11622}, {13223, 13224}, {13250, 13251}, {13254, 13255}, {13263, 13264}, {13284, 13285}, {13290, 13291}, {13302, 13303}, {13306, 13307}, {13308, 13309}, {13315, 13318}, {13316, 13319}, {13317, 13320}, {13718, 13719}, {13841, 13842}, {14337, 14338}, {14339, 14340}, {14406, 23610}, {14610, 32193}, {16156, 16157}, {22733, 22734}, {22888, 22889}, {22933, 22934}, {22984, 22985}, {23035, 23036}, {24809, 24810}, {32312, 32313}, {32407, 32408}, {39904, 39905}, {40282, 40283}, {49628, 49629}

  • X351 = reflection of Xi in Xj for these {i,j}: {2, 11176}, {3, 9126}, {6, 9188}, {669, 8644}, {684, 34291}, {1649, 9125}, {2502, 39527}, {3005, 17414}, {3268, 10190}, {3569, 9208}, {4108, 45317}, {8029, 1637}, {8371, 9189}, {8644, 8651}, {9131, 14610}, {9134, 44564}, {9135, 5027}, {9148, 2}, {9175, 11621}, {9178, 2492}, {9208, 5113}, {9979, 32193}, {11123, 45687}, {11183, 45680}, {11186, 50550}, {11205, 14403}, {11622, 11620}, {13291, 14697}, {14277, 18310}, {14398, 14428}, {14424, 14417}, {17414, 647}, {27798, 45675}, {31986, 32194}, {34290, 45336}, {34291, 6132}, {42663, 9135}, {44420, 39526}

  • X351 = isogonal conjugate of X892

  • X351 = complement of anticomplementary conjugate of X39356

  • X351 = X2-Ceva conjugate of X38988

  • X351 = perspector of hyperbola {A, B, C,X6,X187}}, which is the locus of the barycentric product of circumcircle-X690-antipodes

  • X351 = crosspoint of X110 and X111

  • X351 = crosssum of Xi and Xj for these (i,j): (2, 690), (523, 524), (850, 1236)

  • X351 = crossdifference of every pair of points on line X2X99

  • X351 = circumcircle-inverse of X9129

  • X351 = Parry-isodynamic-circle-inverse of X187

  • X351 = Xi-line conjugate of Xj for these (i,j): {187, 5106}, {690, 2482}, {804, 2}, {2492, 111}, {2793, 9172}, {3455, 41272}, {6131, 7665}, {22105, 7664}

  • X351 = Xi-complementary conjugate of Xj for these (i,j): {31, 38988}, {25322, 21253}.

  • X351 = Xi-Ceva conjugate of Xj for these (i,j): {2, 38988}, {110, 39689}, {111, 3124}, {187, 21906}, {468, 1648}, {598, 35507}, {691, 6}, {827, 6593}, {1177, 3269}, {1383, 38369}, {1576, 47426}, {2489, 21905}, {3455, 20975}, {5467, 187}, {9178, 512}, {10511, 8288}, {14357, 23992}, {14419, 2642}, {22105, 690}, {41498, 14444}, {45773, 20976}

  • X351 = trilinear pole of line {21906, 35507}

  • X351 = Lemoine-axis-intercept of trilinear polar of X512

  • X351 = centroid of the triangle X3X5607X_5608

  • X351 = tripolar centroid of X6

  • X351 = tripolar centroid of X14608

  • X351 = X9227-anticomplementary conjugate of X21294

  • X351 = centroid of Lemoine axis intercepts with sidelines of ABC

  • X351 = X351-of-circumsymmedial-triangle

  • X351 = intersection of tangents to Steiner inellipse at X1084 and X2482

  • X351 = crosspoint wrt medial triangle of X1084 and X2482

  • X351 = vertex conjugate of PU(62)

  • X351 = radical center of (circumcircle, Brocard circle, McCay circumcircle)

  • X351 = harmonic center of circles {X14,X15,X16}} and {X13,X15,X16}}

  • X351 = bicentric difference of PU(i) for these i: 62, 63, 64, 65, 66, 67

  • X351 = PU(62)-harmonic conjugate of X6

  • X351 = PU(63)-harmonic conjugate of X3

  • X351 = PU(64)-harmonic conjugate of X1

  • X351 = PU(65)-harmonic conjugate of X5

  • X351 = PU(66)-harmonic conjugate of X10

  • X351 = PU(67)-harmonic conjugate of X39

  • X351 = X3-of-1st-Parry-triangle

  • X351 = X3-of-2nd-Parry-triangle

  • X351 = X3-of-3rd-Parry-triangle

  • X351 = inverse-in-Parry-isodynamic-circle of X187; see X2

  • X351 = pole of Brocard axis wrt Parry isodynamic circle

  • X351 = bicentric sum of PU(105)

  • X351 = centroid of (degenerate) cross-triangle of 1st and 3rd Parry triangles

  • X351 = centroid of centers of A-, B- and C-Apollonian circles

  • X351 = Lemoine axis (or line PU(2)) intercept of line connecting P(2)-Ceva conjugate of U(2) and U(2)-Ceva conjugate of P(2)

  • X351 = Xi-isoconjugate of Xj for these (i,j): {1, 892}, {2, 36085}, {75, 691}, {76, 36142}, {86, 5380}, {99, 897}, {110, 46277}, {111, 799}, {162, 30786}, {163, 18023}, {561, 32729}, {661, 52940}, {662, 671}, {670, 923}, {811, 895}, {1101, 52632}, {1109, 45773}, {3112, 36827}, {4563, 36128}, {4575, 46111}, {4590, 23894}, {4592, 17983}, {4593, 46154}, {4599, 31125}, {4602, 32740}, {4622, 52747}, {4625, 5547}, {5466, 24041}, {5968, 36036}, {6331, 36060}, {7035, 43926}, {7257, 7316}, {8773, 52035}, {9170, 17955}, {9178, 24037}, {10097, 46254}, {10630, 24039}, {11059, 36045}, {14210, 34574}, {14728, 17467}, {36133, 52756}, {37204, 41272}, {37216, 52141}

  • X351 = Xi-Dao conjugate of Xj for these (i,j): {3, 892}, {115, 18023}, {125, 30786}, {136, 46111}, {206, 691}, {244, 46277}, {512, 9178}, {523, 52632}, {690, 35522}, {1084, 671}, {1560, 6331}, {1648, 3266}, {1649, 850}, {2482, 670}, {2679, 5968}, {3005, 5466}, {3124, 31125}, {5099, 52551}, {5139, 17983}, {6593, 99}, {15477, 34574}, {17413, 42008}, {17423, 895}, {21905, 523}, {21906, 47286}, {23992, 76}, {31654, 11059}, {32664, 36085}, {34452, 36827}, {36830, 52940}, {38986, 897}, {38988, 2}, {38996, 111}, {39010, 52756}, {39072, 52035}, {40368, 32729}, {40600, 5380}, {48317, 264}, {52881, 52608}

  • X351 = barycentric product Xi*Xj for these {i,j}: {1, 2642}, {3, 14273}, {6, 690}, {25, 14417}, {32, 35522}, {37, 14419}, {39, 22105}, {42, 4750}, {50, 51479}, {99, 21906}, {110, 1648}, {111, 1649}, {115, 5467}, {187, 523}, {249, 33919}, {251, 14424}, {468, 647}, {511, 52038}, {512, 524}, {513, 21839}, {525, 44102}, {574, 23287}, {649, 4062}, {661, 896}, {667, 42713}, {669, 3266}, {691, 23992}, {694, 11183}, {798, 14210}, {804, 18872}, {843, 33921}, {850, 14567}, {882, 5026}, {888, 14608}, {922, 1577}, {935, 47415}, {1379, 46462}, {1380, 46463}, {1400, 14432}, {1576, 52628}, {1637, 9717}, {1974, 45807}, {1989, 44814}, {2088, 14559}, {2395, 9155}, {2422, 50567}, {2433, 5642}, {2434, 6791}, {2482, 9178}, {2489, 6390}, {2491, 52145}, {2492, 14357}, {2501, 3292}, {2502, 34763}, {2623, 41586}, {2643, 23889}, {2682, 44769}, {2715, 51429}, {2793, 51927}, {3005, 52898}, {3049, 44146}, {3121, 42721}, {3124, 5468}, {3284, 52475}, {3455, 18311}, {3457, 9204}, {3458, 9205}, {3569, 5967}, {3709, 7181}, {3712, 7180}, {4041, 51653}, {4079, 6629}, {4235, 20975}, {4705, 16702}, {5095, 10097}, {5191, 50942}, {5466, 39689}, {5477, 35364}, {5638, 52723}, {5639, 52722}, {6041, 52094}, {6137, 52039}, {6138, 52040}, {7813, 18105}, {8371, 48450}, {8664, 31068}, {9125, 21448}, {9148, 41309}, {9171, 51226}, {10630, 33915}, {11060, 45808}, {14270, 43084}, {14398, 36890}, {14404, 52757}, {14407, 52759}, {14444, 34574}, {14606, 45330}, {14618, 23200}, {14998, 45662}, {16741, 50487}, {17414, 51541}, {20382, 32583}, {21905, 41909}, {23894, 42081}, {28625, 30595}, {28658, 30605}, {32740, 52629}, {34212, 35282}, {34539, 46049}, {37778, 39201}, {39785, 46001}, {42665, 51823}, {45680, 52660}

  • X351 = barycentric quotient Xi/Xj for these {i,j}: {6, 892}, {31, 36085}, {32, 691}, {110, 52940}, {115, 52632}, {187, 99}, {213, 5380}, {468, 6331}, {512, 671}, {523, 18023}, {524, 670}, {560, 36142}, {647, 30786}, {661, 46277}, {669, 111}, {688, 46154}, {690, 76}, {798, 897}, {887, 14609}, {888, 52756}, {896, 799}, {922, 662}, {1084, 9178}, {1501, 32729}, {1648, 850}, {1649, 3266}, {1692, 52035}, {1924, 923}, {1977, 43926}, {2422, 9154}, {2489, 17983}, {2491, 5968}, {2492, 52551}, {2501, 46111}, {2502, 34760}, {2642, 75}, {2682, 41079}, {3005, 31125}, {3049, 895}, {3051, 36827}, {3124, 5466}, {3266, 4609}, {3292, 4563}, {4062, 1978}, {4750, 310}, {5026, 880}, {5191, 50941}, {5467, 4590}, {5468, 34537}, {5967, 43187}, {6041, 16092}, {6390, 52608}, {6629, 52612}, {8644, 52141}, {9125, 11059}, {9155, 2396}, {9171, 17948}, {9426, 32740}, {9429, 36821}, {9494, 41272}, {11183, 3978}, {14210, 4602}, {14273, 264}, {14398, 9214}, {14407, 52747}, {14417, 305}, {14419, 274}, {14424, 8024}, {14428, 52758}, {14432, 28660}, {14443, 52628}, {14444, 52629}, {14567, 110}, {14608, 886}, {16702, 4623}, {17414, 42008}, {18311, 40074}, {18872, 18829}, {19627, 51478}, {20975, 14977}, {21839, 668}, {21905, 47286}, {21906, 523}, {22105, 308}, {23200, 4558}, {23287, 40826}, {23357, 45773}, {23889, 24037}, {23992, 35522}, {32729, 34539}, {32740, 34574}, {33915, 36792}, {33919, 338}, {33921, 45809}, {35522, 1502}, {39689, 5468}, {41309, 9150}, {42081, 24039}, {42344, 23105}, {42663, 52450}, {42713, 6386}, {44102, 648}, {44814, 7799}, {45680, 41259}, {45807, 40050}, {46001, 18818}, {48450, 9170}, {51479, 20573}, {51653, 4625}, {51927, 46144}, {52038, 290}, {52628, 44173}, {52898, 689}

  • X351 = {Xi,Xj}-harmonic conjugate of Xk for these (i,j,k): {25, 47230, 17994}, {110, 9216, 9145}, {110, 15329, 42743}, {111, 9178, 17993}, {111, 9215, 9142}, {647, 669, 3005}, {647, 8651, 669}, {647, 42659, 42665}, {667, 42653, 42661}, {669, 3005, 8664}, {669, 3231, 42652}, {669, 8665, 3804}, {1649, 14424, 14417}, {2491, 6041, 14398}, {2492, 11631, 17993}, {3569, 5027, 42663}, {3569, 5191, 9409}, {4108, 15724, 45317}, {4455, 21828, 8034}, {5027, 5106, 887}, {5027, 5113, 3569}, {5027, 14270, 5191}, {5029, 5075, 5040}, {5029, 5168, 1960}, {5040, 5163, 890}, {5191, 21731, 42663}, {6137, 6138, 3569}, {6140, 14270, 21731}, {9123, 9131, 14610}, {9123, 9979, 9131}, {9129, 9130, 3}, {9131, 9185, 9979}, {9134, 9189, 44564}, {9134, 44564, 8371}, {9147, 11176, 9148}, {9173, 9174, 8371}, {9185, 9979, 32193}, {11215, 31986, 32194}, {14270, 21731, 9409}, {15724, 36900, 4108}, {42659, 42665, 9409}, {42667, 42668, 669}, {52722, 52723, 11183}

Notes

Let OA be the circle centered at the A-vertex of the 1st anti-Parry triangle and passing through A; define OB and OC cyclically. X351 is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let OA be the circle centered at the A-vertex of the 2nd anti-Parry triangle and passing through A; define OB and OC cyclically. X351 is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

Let S be the set of these 9 circles: circumcircle, Brocard circle, ninepoint circle of tangential triangle, McCay circumcircle, Lucas inner circle, Lucas circles radical circle, outer Montesdeoca-Lemoine circle, inner Montesdeoca-Lemoine circle, Parry isodynamic circle. Let S' be any one of the 84 subsets of S that consist of 3 circles. The radical center of S' is X351. (Peter Moses, March 4, 2023)

Taken in pairs (i.e. {{S', radical axis of S'}…}, we have the following radical axes:

{{circumcircle, Brocard circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{circumcircle, ninepoint circle of tangential triangle},{},{110, 351, 526, 684, 1576, 1624, 1634, 2421, 4556, 4558, …}}

{{circumcircle, McCay circumcircle},{},{351}}

{{circumcircle, Lucas inner circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{circumcircle, Lucas circles radical circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{circumcircle, outer Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{circumcircle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{circumcircle, Parry isodynamic circle},{},{6, 351, 9023, 9188, 48450}}

{{Brocard circle, ninepoint circle of tangential triangle},{},{351, 3265}}

{{Brocard circle, McCay circumcircle},{},{351, 10166, 11171}}

{{Brocard circle, Lucas inner circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Brocard circle, Lucas circles radical circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Brocard circle, outer Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Brocard circle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Brocard circle, Parry isodynamic circle},{},{351, 574}}

{{ninepoint circle of tangential triangle, McCay circumcircle},{},{351}}

{{ninepoint circle of tangential triangle, Lucas inner circle},{},{351}}

{{ninepoint circle of tangential triangle, Lucas circles radical circle},{},{351}}

{{ninepoint circle of tangential triangle, outer Montesdeoca-Lemoine circle},{},{351}}

{{ninepoint circle of tangential triangle, inner Montesdeoca-Lemoine circle},{},{351}}

{{ninepoint circle of tangential triangle, Parry isodynamic circle},{},{351}}

{{McCay circumcircle, Lucas inner circle},{},{351}}

{{McCay circumcircle, Lucas circles radical circle},{},{351}}

{{McCay circumcircle, outer Montesdeoca-Lemoine circle},{},{351}}

{{McCay circumcircle, inner Montesdeoca-Lemoine circle},{},{351}}

{{McCay circumcircle, Parry isodynamic circle},{},{351}}

{{Lucas inner circle, Lucas circles radical circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Lucas inner circle, outer Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Lucas inner circle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Lucas inner circle, Parry isodynamic circle},{},{351, 6468}}

{{Lucas circles radical circle, outer Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Lucas circles radical circle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{Lucas circles radical circle, Parry isodynamic circle},{},{351, 6221}}

{{outer Montesdeoca-Lemoine circle, inner Montesdeoca-Lemoine circle},{Lemoine axis},{187, 237, 351, 512, 647, 649, 663, 665, 667, 669, …}}

{{outer Montesdeoca-Lemoine circle, Parry isodynamic circle},{},{351, 50663}}

{{inner Montesdeoca-Lemoine circle, Parry isodynamic circle},{},{351, 50662}} (Peter Moses, March 4, 2023)