X₁₁₁

PARRY POINT

X₁₁₁ = anticomplement of X₁₂₆

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

X₁₁₁ (PARRY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1296, 3}

• Related Centers: X₁₂₉₆ (COLLINGS TRANSFORM OF X₁₂₆), X₃ (CIRCUMCENTER)

Trilinears: \(a/(2a^{2} - b^{2} - c^{2}) : b/(2b^{2} - c^{2} - a^{2}) : c/(2c^{2} - a^{2} - b^{2})\)

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

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

Search Value (6, 9, 13): \(1.207441876625\)

X₁₁₁ lies on the Parry circle.

▸ Incident Central Lines305 lines

{1, 8691}, {2, 99}, {3, 1296}, {4, 1560}, {5, 10748}, {6, 110}, {11, 10779}, {15, 9202}, {16, 9203}, {20, 20187}, {22, 2079}, {23, 187}, {24, 8428}, {25, 112}, {30, 2696}, {32, 1383}, {37, 100}, {39, 7711}, {42, 101}, {51, 10558}, {55, 6019}, {56, 3325}, {74, 2433}, {76, 16055}, {98, 1637}, {102, 2819}, {103, 2824}, {104, 2830}, {105, 2837}, {106, 2843}, {107, 393}, {108, 1880}, {109, 1400}, {125, 35902}, {140, 38806}, {141, 36883}, {182, 353}, {183, 9066}, {184, 3048}, {186, 10098}, {230, 476}, {232, 1304}, {251, 827}, {263, 9463}, {265, 14846}, {308, 689}, {316, 35569}, {323, 2987}, {325, 2858}, {351, 2492}, {352, 511}, {376, 30256}, {381, 6032}, {382, 38799}, {385, 892}, {468, 935}, {477, 5915}, {493, 1306}, {494, 1307}, {512, 843}, {513, 2721}, {514, 2729}, {515, 2735}, {516, 2740}, {517, 2746}, {518, 2753}, {519, 2759}, {520, 2763}, {521, 2767}, {522, 2768}, {523, 2770}, {524, 6082}, {526, 9184}, {538, 5971}, {542, 6791}, {548, 38798}, {550, 38797}, {576, 8600}, {612, 15322}, {631, 38804}, {647, 842}, {649, 2712}, {650, 2752}, {667, 35107}, {669, 5970}, {675, 26273}, {694, 805}, {729, 5027}, {739, 5040}, {741, 3572}, {755, 5113}, {759, 23894}, {831, 1961}, {840, 5098}, {850, 2868}, {858, 24855}, {880, 34087}, {898, 5291}, {901, 17735}, {902, 2054}, {907, 1180}, {919, 20989}, {925, 2165}, {927, 37798}, {930, 2963}, {931, 941}, {932, 16606}, {933, 8882}, {934, 1427}, {1055, 2701}, {1084, 14948}, {1113, 8106}, {1114, 8105}, {1171, 6578}, {1194, 3108}, {1239, 6572}, {1241, 35567}, {1287, 10416}, {1289, 6353}, {1290, 3290}, {1291, 14579}, {1292, 4220}, {1293, 9526}, {1294, 9529}, {1295, 9531}, {1297, 13114}, {1302, 3018}, {1305, 26252}, {1310, 17019}, {1340, 6141}, {1341, 6142}, {1344, 8427}, {1345, 8426}, {1350, 33979}, {1379, 5639}, {1380, 5638}, {1495, 1976}, {1499, 6093}, {1611, 36616}, {1640, 14932}, {1645, 9468}, {1648, 9140}, {1657, 38800}, {1691, 32694}, {1692, 35265}, {1799, 6573}, {1992, 10552}, {2080, 13225}, {2177, 5147}, {2248, 33774}, {2291, 5075}, {2373, 9979}, {2374, 9131}, {2378, 6138}, {2379, 6137}, {2393, 5166}, {2408, 9084}, {2444, 9135}, {2469, 14899}, {2470, 35607}, {2690, 3011}, {2691, 37959}, {2697, 6587}, {2698, 3288}, {2703, 3230}, {2713, 3331}, {2748, 5524}, {2760, 3239}, {2766, 5089}, {2782, 9775}, {2981, 10409}, {2998, 3222}, {3003, 9060}, {3016, 9160}, {3055, 10276}, {3060, 10559}, {3090, 38807}, {3098, 8617}, {3143, 34171}, {3146, 38801}, {3163, 5304}, {3430, 28295}, {3448, 6388}, {3457, 5995}, {3458, 5994}, {3522, 38803}, {3542, 30251}, {3815, 30537}, {3832, 38802}, {3920, 9281}, {4231, 26706}, {4240, 6531}, {4563, 25047}, {4588, 28658}, {5012, 10485}, {5024, 9155}, {5033, 7712}, {5038, 5643}, {5085, 14688}, {5092, 14660}, {5099, 30718}, {5108, 5969}, {5169, 18424}, {5182, 35356}, {5251, 29351}, {5468, 10754}, {5475, 11640}, {5477, 9143}, {5622, 35901}, {5916, 9200}, {5917, 9201}, {6034, 7698}, {6037, 9154}, {6083, 20461}, {6094, 9080}, {6128, 7736}, {6151, 10410}, {6200, 7598}, {6221, 7601}, {6323, 9208}, {6325, 9769}, {6339, 20080}, {6396, 7599}, {6398, 7602}, {6781, 37901}, {7467, 14678}, {7485, 34866}, {7492, 8588}, {7496, 8589}, {7575, 32229}, {7606, 10166}, {7610, 9829}, {7792, 9069}, {7954, 14075}, {8030, 38020}, {8429, 37930}, {8586, 9225}, {8644, 9136}, {8651, 14659}, {8652, 28625}, {8681, 15387}, {8707, 14624}, {8709, 27809}, {8744, 10422}, {8859, 18818}, {8869, 19121}, {9059, 26244}, {9067, 26243}, {9070, 26242}, {9076, 22105}, {9091, 16098}, {9096, 26241}, {9126, 19901}, {9137, 10102}, {9159, 9828}, {9169, 9830}, {9176, 13233}, {9189, 18007}, {9206, 11081}, {9207, 11086}, {9605, 30734}, {9810, 14126}, {9831, 12434}, {10304, 37749}, {10313, 10420}, {10560, 11002}, {10645, 14704}, {10646, 14705}, {11175, 36213}, {11215, 11622}, {11332, 21444}, {11632, 14694}, {11635, 38463}, {11639, 31489}, {11645, 11647}, {12041, 35447}, {12149, 25424}, {12212, 31609}, {12834, 14153}, {13397, 26253}, {13398, 26283}, {14733, 17966}, {15018, 30535}, {15066, 35575}, {15271, 34227}, {15360, 15993}, {16081, 22456}, {16167, 16310}, {16318, 22239}, {16806, 21461}, {16807, 21462}, {19561, 28895}, {21906, 34574}, {23297, 33013}, {23701, 30209}, {30249, 33630}, {31884, 37751}, {32237, 38010}, {32526, 35002}, {32849, 35574}, {34079, 36069}, {34482, 34572}, {36066, 36085}, {36201, 36203}, {36894, 37643}

▸ Functional Relations55 relations

• X₁₁₁ = reflection of X₁₂₉₆ in X₃

• X₁₁₁ = isogonal conjugate of X₅₂₄

• X₁₁₁ = isotomic conjugate of X₃₂₆₆

• X₁₁₁ = inverse-in-Brocard-circle of X₃₅₃

• X₁₁₁ = anticomplement of X₁₂₆

• X₁₁₁ = cevapoint of X₆ and X₁₈₇

• X₁₁₁ = X_i-cross conjugate of X_j for these (i,j): (23, 251), (187, 6), (351, 110)

• X₁₁₁ = crossdifference of every pair of points on line X₃₅₁X₆₉₀

• X₁₁₁ = perspector of ABC and the triangle formed by the reflections of line PU(7) in the sides of ABC

• X₁₁₁ = point of intersection, other than A, B, C, of circumcircle and hyperbola {A, B, C,X₂,X₆}}

• X₁₁₁ = trilinear pole of line X₆X₅₁₂

• X₁₁₁ = Λ(X₂, X₆)

• X₁₁₁ = Ψ(X_i,X_j) for these (i,j): (6, 512), (21, 650), (190, 10)

• X₁₁₁ = trilinear pole wrt circumsymmedial triangle of Brocard axis

• X₁₁₁ = trilinear pole wrt circummedial triangle of line X₂X₆

• X₁₁₁ = X₁₁₀-of-circumsymmedial-triangle

• X₁₁₁ = X₂₃-of-4th-Brocard-triangle

• X₁₁₁ = X₃₅₂-of-orthocentroidal-triangle

• X₁₁₁ = X₁₃₈-of-excentral-triangle

• X₁₁₁ = reflection of X₂₇₇₀ in the Euler line

• X₁₁₁ = reflection of X₈₄₃ in the Brocard axis

• X₁₁₁ = reflection of X₂₇₂₁ in line X₁X₃

• X₁₁₁ = reflection of X₁₁₀ in line X₃X₃₅₁

• X₁₁₁ = inverse-in-polar-circle of X₁₅₆₀

• X₁₁₁ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₁₅

• X₁₁₁ = inverse-in-Moses-radical-circle of X₈₄₂

• X₁₁₁ = inverse-in-circle-O(15,16) of X₆₉₁

• X₁₁₁ = X₁₅₇₇-isoconjugate of X₅₄₆₇

• X₁₁₁ = SR(P,U), where P and U are the circumcircle intercepts of line X₂X₆

• X₁₁₁ = one of two harmonic traces of the McCay circles; X₂ is the other

• X₁₁₁ = X₁₂₉₆-of-1st-Parry-triangle

• X₁₁₁ = X₁₁₁-of-2nd-Parry-triangle

• X₁₁₁ = X₆₉₁-of-3rd-Parry-triangle

• X₁₁₁ = center of similitude of ABC and 2nd Parry triangle

• X₁₁₁ = inverse-in-Parry-isodynamic-circle of X₁₁₀; see X₂

• X₁₁₁ = 3rd-Parry-to-circumsymmedial similarity image of X₃₅₂

• X₁₁₁ = center of similitude of ABC and circumsymmedial triangle of Artzt triangle

• X₁₁₁ = eigencenter of circumtangential triangle

• X₁₁₁ = perspector of ABC and unary cofactor triangle of 2nd Brocard triangle

• X₁₁₁ = X₂-of-4th-anti-Brocard-triangle

• X₁₁₁ = anti-Artzt-to-4th-anti-Brocard similarity image of X₂

• X₁₁₁ = Thomson-isogonal conjugate of X₁₄₉₉

• X₁₁₁ = Lucas-isogonal conjugate of X₁₄₉₉

• X₁₁₁ = Cundy-Parry Phi transform of X₁₄₂₆₂

• X₁₁₁ = Cundy-Parry Psi transform of X₁₃₆₀₈

• X₁₁₁ = X₆-isoconjugate of X₁₄₂₁₀

• X₁₁₁ = inverse-in-circle-{X₁₆₈₇,X₁₆₈₈,PU(1),PU(2)}} of X₃₂₆₉₄

• X₁₁₁ = barycentric product of circumcircle intercepts of line X₂X₅₂₃

• X₁₁₁ = inverse-in-Stevanovic-circle of X₂₇₅₈

• X₁₁₁ = trilinear pole, wrt Thomson triangle, of line X₆X₅₆₄₆

• X₁₁₁ = trilinear pole, wrt 2nd Parry triangle, of line X₅₁₁X₃₅₆₉

• X₁₁₁ = areal center of pedal triangles of PU(7)

• X₁₁₁ = X₂-Ceva conjugate of X₁₅₈₉₉

• X₁₁₁ = perspector of circumconic centered at X₁₅₈₉₉

• X₁₁₁ = McCay-bisector-circle-inverse of X₂

Notes

Let L be a line tangent to the Brocard circle. Let P be the trilinear pole of L, and let P' be the isogonal conjugate of P. As L varies, P' traces a parabola with focus at X₁₁₁. The parabola meets the line at infinity at X₅₂₄. Also, X₁₁₁ is the QA-P₄ center (Isogonal Center) of quadrangle X₁₃X₁₄X₁₅X₁₆ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html) Also, let A' = BC∩X₁₁₅X₁₂₅, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₁₁. (Randy Hutson, October 13, 2015)

Let A″B″C″ be the 2nd Ehrmann triangle. Let P_a be the pole of line B″C″ wrt the A-Ehrmann circle, and define P_b and P_c cyclically. Let P_a' be the pole of line BC wrt the A-Ehrmann circle, and define P_b' and P_c' cyclically. The lines APaPa’, BPbPb’, CPcPc’ concur in X₁₁₁. Also, let A* be the trilinear pole of line B″C″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X₁₁₁. (Randy Hutson, November 18, 2015)

Let A₁B₁C₁ and A₃B₃C₃ be the 1st and 3rd Parry triangles. Let A' be the barycentric product A₁*A₃, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₁₁. (Randy Hutson, February 10, 2016)

Let A'B'C' and A″B″C″ be the 4th Brocard and 4th anti-Brocard triangles, resp. Let A* be the barycentric product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₁₁₁. (Randy Hutson, December 2, 2017)