X₂₁

SCHIFFLER POINT

Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X₂₁. This configuration extends to Kirikami-Schiffler points and generalizations found by Peter Moses, as introduced just before X₃₆₄₈.

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

X₂₁ (SCHIFFLER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {1, 31}

• Related Centers: X₁ (INCENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₃₁ (2nd POWER POINT)

Trilinears: \(1/(\cos B + \cos C) : 1/(\cos C + \cos A) : 1/(\cos A + \cos B)\)

Barycentrics: \(a/(\cos B + \cos C) : b/(\cos C + \cos A) : c/(\cos A + \cos B)\)

Search Value (6, 9, 13): \(3.177166860315\)

▸ Alternate Coordinates1 form

• Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (b + c - a)/(b + c)\)

▸ Linear Combinations1 combination

• X₂₁ = \(3R\) X₂ + \(2r\) X₃

▸ Incident Central Lines782 lines

{1, 31}, {2, 3}, {6, 941}, {7, 56}, {8, 55}, {9, 41}, {10, 35}, {11, 4996}, {12, 5080}, {15, 5362}, {16, 5367}, {19, 4288}, {32, 981}, {34, 17080}, {36, 79}, {37, 172}, {39, 33854}, {40, 3577}, {42, 4281}, {43, 37574}, {44, 4273}, {45, 3285}, {46, 17098}, {48, 15656}, {51, 970}, {57, 4652}, {60, 960}, {65, 4640}, {69, 10432}, {71, 4269}, {72, 943}, {73, 651}, {74, 34800}, {75, 272}, {76, 37670}, {77, 1394}, {84, 285}, {85, 3188}, {90, 224}, {92, 41227}, {99, 105}, {101, 3294}, {104, 110}, {106, 34594}, {107, 1295}, {108, 39435}, {109, 37558}, {112, 26703}, {119, 31659}, {125, 38612}, {141, 4265}, {142, 10123}, {144, 954}, {145, 956}, {149, 2894}, {162, 3194}, {177, 7587}, {183, 18135}, {184, 13323}, {187, 5277}, {193, 37492}, {194, 16998}, {198, 5296}, {200, 4866}, {210, 4420}, {214, 501}, {219, 2335}, {220, 40779}, {226, 37583}, {238, 256}, {243, 1896}, {261, 314}, {268, 280}, {270, 1172}, {286, 1441}, {294, 1212}, {307, 2062}, {321, 7283}, {323, 5453}, {329, 5703}, {332, 1036}, {355, 11491}, {385, 1655}, {386, 1724}, {387, 24597}, {388, 37579}, {390, 6601}, {391, 4254}, {394, 36746}, {476, 2687}, {484, 3754}, {495, 20060}, {497, 10527}, {498, 11681}, {499, 14793}, {511, 15988}, {514, 23775}, {515, 10902}, {516, 15909}, {517, 1389}, {518, 2346}, {519, 3746}, {527, 34917}, {529, 15888}, {535, 5270}, {551, 5557}, {572, 1765}, {593, 6051}, {603, 17074}, {612, 989}, {614, 988}, {643, 1320}, {644, 1334}, {645, 36798}, {662, 1156}, {667, 29150}, {672, 41239}, {691, 2752}, {740, 27368}, {741, 932}, {748, 978}, {750, 37603}, {756, 5293}, {757, 15569}, {849, 30576}, {884, 885}, {900, 42741}, {902, 5255}, {908, 12572}, {912, 24299}, {915, 925}, {930, 26707}, {934, 26702}, {936, 3305}, {938, 5744}, {940, 4252}, {942, 3218}, {944, 10267}, {946, 11012}, {950, 5745}, {952, 34352}, {961, 1402}, {962, 3428}, {965, 37504}, {966, 36744}, {976, 983}, {982, 28082}, {986, 3924}, {987, 2206}, {990, 9962}, {991, 37659}, {992, 5110}, {995, 15315}, {999, 3296}, {1019, 35355}, {1021, 23893}, {1030, 1213}, {1038, 1041}, {1039, 1040}, {1056, 10587}, {1058, 10529}, {1060, 1063}, {1061, 1062}, {1064, 3073}, {1078, 18140}, {1083, 3110}, {1104, 3666}, {1107, 1914}, {1149, 8421}, {1150, 10449}, {1152, 31473}, {1155, 3812}, {1183, 7058}, {1191, 40153}, {1201, 17187}, {1210, 41557}, {1211, 26064}, {1214, 1396}, {1220, 26115}, {1251, 5240}, {1254, 1758}, {1260, 20007}, {1261, 4723}, {1279, 16696}, {1290, 12030}, {1296, 9061}, {1304, 2694}, {1319, 1408}, {1329, 5432}, {1330, 3936}, {1335, 9678}, {1376, 5217}, {1392, 2098}, {1412, 1420}, {1414, 43736}, {1453, 5256}, {1466, 5435}, {1470, 5555}, {1475, 16503}, {1478, 10198}, {1479, 11680}, {1482, 14497}, {1500, 5291}, {1503, 26543}, {1610, 2217}, {1617, 3600}, {1626, 40462}, {1633, 24723}, {1682, 3271}, {1697, 3680}, {1698, 5010}, {1706, 35445}, {1709, 9961}, {1728, 41568}, {1761, 2294}, {1762, 18673}, {1764, 10451}, {1770, 12609}, {1781, 25081}, {1788, 11509}, {1798, 40454}, {1805, 30556}, {1806, 7133}, {1807, 35194}, {1808, 43748}, {1809, 36795}, {1834, 24883}, {1836, 28628}, {1837, 26066}, {1870, 37565}, {1936, 2654}, {1946, 4391}, {1975, 16992}, {1993, 36742}, {1994, 36750}, {2053, 7155}, {2077, 6684}, {2078, 10106}, {2096, 5553}, {2112, 39244}, {2136, 31509}, {2175, 35628}, {2183, 41263}, {2223, 16830}, {2238, 18755}, {2241, 16975}, {2269, 23640}, {2271, 37657}, {2274, 16690}, {2276, 4426}, {2280, 21384}, {2295, 17735}, {2310, 2648}, {2320, 5289}, {2341, 5549}, {2344, 3061}, {2417, 43737}, {2550, 7676}, {2551, 5218}, {2778, 12826}, {2782, 5985}, {2787, 16158}, {2802, 13143}, {2804, 14224}, {2886, 6284}, {2895, 41014}, {2906, 18455}, {2979, 37482}, {3006, 5015}, {3011, 13161}, {3035, 17100}, {3052, 5710}, {3053, 5275}, {3058, 3813}, {3060, 5752}, {3062, 5732}, {3074, 22350}, {3085, 3436}, {3086, 8071}, {3120, 24161}, {3178, 38456}, {3207, 5781}, {3208, 4390}, {3210, 19851}, {3216, 4256}, {3220, 4357}, {3241, 3303}, {3244, 5288}, {3254, 12053}, {3256, 4848}, {3270, 43746}, {3290, 16716}, {3293, 33771}, {3304, 5558}, {3306, 15803}, {3315, 3953}, {3333, 4666}, {3336, 5883}, {3337, 4973}, {3361, 10582}, {3419, 5791}, {3421, 10528}, {3423, 17206}, {3427, 5731}, {3434, 4294}, {3452, 27385}, {3453, 30115}, {3454, 25645}, {3467, 10176}, {3474, 28629}, {3476, 11510}, {3487, 5905}, {3488, 12649}, {3496, 5060}, {3501, 41423}, {3509, 21808}, {3551, 15485}, {3555, 3957}, {3565, 15344}, {3579, 3753}, {3583, 14794}, {3585, 3822}, {3586, 5705}, {3589, 5096}, {3614, 6668}, {3617, 5687}, {3618, 36741}, {3621, 7317}, {3623, 6767}, {3624, 5561}, {3632, 25439}, {3634, 9342}, {3635, 13602}, {3663, 17189}, {3670, 30117}, {3673, 16749}, {3678, 7161}, {3679, 8715}, {3681, 3811}, {3684, 3691}, {3689, 4662}, {3695, 32849}, {3701, 7081}, {3716, 8648}, {3720, 37607}, {3730, 14964}, {3731, 33628}, {3733, 23836}, {3737, 6615}, {3738, 35055}, {3742, 32636}, {3748, 34791}, {3757, 4968}, {3816, 5433}, {3825, 10090}, {3826, 26060}, {3827, 41582}, {3831, 32918}, {3833, 5131}, {3841, 4324}, {3870, 7160}, {3886, 4483}, {3895, 4853}, {3912, 24632}, {3920, 5266}, {3925, 15338}, {3928, 11518}, {3929, 3951}, {3931, 17016}, {3935, 34790}, {3948, 26243}, {3971, 8669}, {4005, 15481}, {4011, 25591}, {4026, 20872}, {4067, 41696}, {4084, 5425}, {4101, 4416}, {4129, 39577}, {4251, 16552}, {4253, 16783}, {4255, 4383}, {4257, 37522}, {4258, 37658}, {4293, 7742}, {4297, 12617}, {4302, 19854}, {4309, 31458}, {4314, 4847}, {4322, 9363}, {4326, 42015}, {4344, 21002}, {4359, 16817}, {4366, 26801}, {4385, 26227}, {4413, 19877}, {4418, 24850}, {4422, 30906}, {4423, 5204}, {4427, 17164}, {4436, 40625}, {4438, 36568}, {4482, 29699}, {4516, 4612}, {4518, 18265}, {4520, 6603}, {4558, 8759}, {4565, 9372}, {4567, 5377}, {4570, 24433}, {4646, 4689}, {4668, 4803}, {4679, 25681}, {4680, 30172}, {4857, 10707}, {4867, 5424}, {4881, 10308}, {4972, 34868}, {4995, 21031}, {5016, 33113}, {5044, 5440}, {5045, 29817}, {5082, 20075}, {5119, 14923}, {5124, 17398}, {5132, 17277}, {5138, 10477}, {5176, 10039}, {5179, 27068}, {5183, 10107}, {5211, 31108}, {5239, 33653}, {5263, 8053}, {5264, 30116}, {5278, 9534}, {5280, 25092}, {5281, 7080}, {5287, 37554}, {5294, 5314}, {5297, 37589}, {5300, 29641}, {5325, 12437}, {5414, 31453}, {5422, 36754}, {5438, 7308}, {5439, 27003}, {5444, 14800}, {5482, 33852}, {5484, 20999}, {5506, 13146}, {5535, 31870}, {5537, 43174}, {5554, 5657}, {5584, 9778}, {5587, 6796}, {5603, 11249}, {5686, 6600}, {5690, 11849}, {5692, 9275}, {5694, 37733}, {5697, 21398}, {5708, 23958}, {5709, 21165}, {5711, 17126}, {5719, 17484}, {5735, 11522}, {5777, 33597}, {5790, 32141}, {5795, 6735}, {5814, 33077}, {5818, 11499}, {5832, 12701}, {5836, 37568}, {5880, 30295}, {5882, 10031}, {5886, 16159}, {5901, 22765}, {5902, 15173}, {5903, 30147}, {5919, 11260}, {5943, 15489}, {6001, 23059}, {6043, 17015}, {6147, 17483}, {6211, 25024}, {6326, 20117}, {6361, 35239}, {6514, 30223}, {6516, 17095}, {6554, 32561}, {6595, 17643}, {6599, 21634}, {6626, 7261}, {6651, 27954}, {6667, 7294}, {6693, 25441}, {6713, 18861}, {6727, 15997}, {6737, 18249}, {6745, 18250}, {6762, 10389}, {7004, 40602}, {7049, 7361}, {7085, 26065}, {7100, 13486}, {7149, 8885}, {7173, 31260}, {7179, 25581}, {7191, 37592}, {7226, 36565}, {7253, 23189}, {7257, 8851}, {7259, 30618}, {7270, 33116}, {7284, 37618}, {7292, 37599}, {7330, 12528}, {7354, 25466}, {7373, 18490}, {7588, 8250}, {7595, 8225}, {7621, 32479}, {7680, 11827}, {7688, 31730}, {7713, 24611}, {7745, 37661}, {7750, 37664}, {7754, 17002}, {7783, 17000}, {7793, 16997}, {7967, 16202}, {8062, 23226}, {8109, 8391}, {8110, 8372}, {8185, 39582}, {8227, 16125}, {8273, 10429}, {8296, 16484}, {8568, 34867}, {8582, 10164}, {8686, 8690}, {8720, 24165}, {8760, 26641}, {8844, 33295}, {8847, 43747}, {8886, 41084}, {8932, 43751}, {8983, 19080}, {9579, 25525}, {9612, 31266}, {9656, 34739}, {9668, 31493}, {9670, 11235}, {9710, 34612}, {9956, 33862}, {9957, 38460}, {10085, 12669}, {10087, 12531}, {10157, 40262}, {10165, 12608}, {10167, 34862}, {10179, 20323}, {10197, 37719}, {10202, 26877}, {10246, 13465}, {10266, 12524}, {10305, 24558}, {10307, 38031}, {10309, 24556}, {10385, 43745}, {10390, 18164}, {10396, 41576}, {10435, 10444}, {10436, 18655}, {10454, 13478}, {10544, 40966}, {10571, 34027}, {10585, 10590}, {10595, 10680}, {10601, 36745}, {10679, 12245}, {10914, 24297}, {11231, 17619}, {11240, 42842}, {11374, 31053}, {11375, 18977}, {11376, 16142}, {11507, 18391}, {11544, 20084}, {11683, 25255}, {11752, 15788}, {11789, 15789}, {12388, 12390}, {12515, 13145}, {12522, 12538}, {12523, 12539}, {12527, 13405}, {12532, 12739}, {12607, 34606}, {12615, 15326}, {12641, 13278}, {12671, 37837}, {12699, 17173}, {12775, 37562}, {12913, 37737}, {12953, 31245}, {13100, 15325}, {13151, 13369}, {13205, 32157}, {13384, 15829}, {13397, 39439}, {13464, 34485}, {13887, 19014}, {13940, 19013}, {13971, 19079}, {14192, 37741}, {14496, 31663}, {14526, 16152}, {14547, 41243}, {14795, 37710}, {14803, 16154}, {14804, 37701}, {14882, 40663}, {15178, 37518}, {15179, 24928}, {15446, 17104}, {15654, 24552}, {15852, 25939}, {16020, 16752}, {16114, 27180}, {16153, 33593}, {16155, 30384}, {16466, 17127}, {16478, 17017}, {16502, 31449}, {16549, 24047}, {16566, 32118}, {16579, 33178}, {16678, 23383}, {16683, 16693}, {16684, 32922}, {16691, 23393}, {16712, 33955}, {16720, 24358}, {16780, 31429}, {16824, 32932}, {16826, 37609}, {16887, 17219}, {16974, 41269}, {16996, 20081}, {17019, 33774}, {17052, 25447}, {17054, 17595}, {17056, 24936}, {17123, 27627}, {17168, 41691}, {17171, 18589}, {17181, 27187}, {17202, 29097}, {17257, 24320}, {17349, 37502}, {17379, 37507}, {17496, 22160}, {17594, 25059}, {17596, 24443}, {17601, 24440}, {17609, 42819}, {17613, 31787}, {17733, 32915}, {17749, 37687}, {17756, 31448}, {17778, 20077}, {17811, 37501}, {18123, 34435}, {18228, 27383}, {18299, 31008}, {18357, 18524}, {18481, 22798}, {18623, 41402}, {18642, 26167}, {18645, 43177}, {18653, 24564}, {18990, 20067}, {19684, 19762}, {19701, 19759}, {19716, 19753}, {19732, 19760}, {19783, 44094}, {19784, 37557}, {19785, 19844}, {19786, 19841}, {19808, 19842}, {19822, 19845}, {19862, 25542}, {20018, 37652}, {20066, 31419}, {20470, 25508}, {20653, 33160}, {21044, 23907}, {21246, 27401}, {21321, 40605}, {22080, 35203}, {22344, 24627}, {22345, 38000}, {22369, 26045}, {22376, 27002}, {22753, 38306}, {22768, 31631}, {22935, 36865}, {23144, 34046}, {23181, 35097}, {23206, 26634}, {23369, 23843}, {23537, 33129}, {23864, 27527}, {24159, 33146}, {24392, 41864}, {24436, 24697}, {24467, 37615}, {24470, 26842}, {24512, 33863}, {24586, 29966}, {24602, 29968}, {24619, 26526}, {24700, 25371}, {24880, 31204}, {24931, 31247}, {25005, 26285}, {25055, 25056}, {25354, 34053}, {25500, 30949}, {26102, 37608}, {26128, 36505}, {26144, 39200}, {26241, 39581}, {26244, 27040}, {26321, 34773}, {26437, 41545}, {26487, 37821}, {26558, 26629}, {26728, 26729}, {26818, 42884}, {26878, 31837}, {26921, 37533}, {26932, 43735}, {27006, 34847}, {27025, 31020}, {27097, 27185}, {27804, 41813}, {28386, 41346}, {28612, 34886}, {28813, 30847}, {29632, 30984}, {30302, 30387}, {30303, 30388}, {30304, 30363}, {31157, 37722}, {31393, 36846}, {31546, 31549}, {32010, 40415}, {32456, 36812}, {32919, 35633}, {32937, 36507}, {33297, 34016}, {33668, 37535}, {33814, 34122}, {33860, 34123}, {33925, 34610}, {34124, 38619}, {34277, 39167}, {34545, 37509}, {35658, 35660}, {36607, 38249}, {37503, 37654}, {37605, 41695}, {38722, 38752}, {38859, 40719}, {40457, 41364}, {41592, 41728}, {41601, 41734}, {41604, 41739}, {41606, 41741}

▸ Functional Relations31 relations

• X₂₁ = midpoint of X₁ and X₁₉₁

• X₂₁ = reflection of X₃₆₅₁ in X₃

• X₂₁ = isogonal conjugate of X₆₅

• X₂₁ = isotomic conjugate of X₁₄₄₁

• X₂₁ = circumcircle-inverse of X₁₃₂₅

• X₂₁ = polar-circle-inverse of X₃₇₉₈₂

• X₂₁ = orthoptic-circle-of-Steiner-inellipse-inverse of complement of X₃₇₉₅₉

• X₂₁ = orthoptic-circle-of-Steiner-circumellipse-inverse of anticomplement of X₃₇₉₅₉

• X₂₁ = complement of X₂₄₇₅

• X₂₁ = anticomplement of X₄₄₂

• X₂₁ = X_i-Ceva conjugate of X_j for these (i,j): (86, 81), (261, 333)

• X₂₁ = cevapoint of X_i and X_j for these (i,j): (1, 3), (9, 55), (1805, 1806)

• X₂₁ = X_i-cross conjugate of X_j for these (i,j): (1, 29), (3, 283), (9, 333), (55, 284), (58, 285), (284, 81), (522, 100)

• X₂₁ = crosspoint of X_i and X_j for these {i,j}: {86, 333}, {1805, 1806}

• X₂₁ = crosssum of X_i and X_j for these (i,j): (1, 1046), (42, 1400), (1254, 1425), (1402, 1409)

• X₂₁ = crossdifference of every pair of points on line X₆₄₇X₆₆₁

• X₂₁ = X_i-Hirst inverse of X_j for these (i,j): (2, 448), (3, 416), (4, 425)

• X₂₁ = X_i-beth conjugate of X_j for these (i,j): (21, 58), (99, 21), (643, 21), (1043, 1043), (1098, 21)

• X₂₁ = intersection of tangents at X₁ and X₃ to the Stammler hyperbola

• X₂₁ = X₅₄-of-2nd-circumperp-triangle

• X₂₁ = X₃₅₇₄-of-excentral-triangle

• X₂₁ = crosspoint of X₁ and X₃ wrt the excentral triangle

• X₂₁ = crosspoint of X₁ and X₃ wrt the tangential triangle

• X₂₁ = trilinear pole of line X₅₂₁X₆₅₀

• X₂₁ = similitude center of ABC and X₁-Brocard triangle

• X₂₁ = X_i-isoconjugate of X_j for these (i,j): (6, 226), (75, 1402)

• X₂₁ = {X₁,X₆₃}-harmonic conjugate of X₃₈₆₈

• X₂₁ = perspector of 2nd circumperp triangle and cross-triangle of ABC and 2nd circumperp triangle

• X₂₁ = perspector of ABC and cross-triangle of ABC and 1st Conway triangle

• X₂₁ = perspector of Gemini triangles 1 and 8

• X₂₁ = barycentric product of Feuerbach hyperbola intercepts of line X₂X₆

Notes

As a point on the Euler line, X₂₁ has Shinagawa coefficients \((\$aS_{A}\$, abc - \$aS_{A}\$)\).

The name of this point honors Kurt Schiffler (../bstud/schiffler.html).

Let I_aI_bI_c be the incentral triangle of ABC, and let L_A be the reflection of line I_bI_c in line BC; define L_B and L_C cyclically. The triangle formed by the lines L_A, L_B, L_C is perspective to ABC, and the perspector is X₂₁. (Randy Hutson, 9/23/2011)

Let A'B'C' be the 2nd circumperp triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, and CC″ concur in X₂₁. (Randy Hutson, April 9, 2016)

Let Eu_aEu_bEu_c be the Feuerbach triangle. Let L_a be the tangent to the nine-point circle at Eu_a, and define L_b and L_c cyclically. Let A″ be the isogonal conjugate of the trilinear pole of L_a, and define B″ and C″ cyclically. Let A*B*C* be the circumcevian triangle, wrt A″B″C″, of X₁. The lines AA*, BB*, CC* concur in X₂₁. (Randy Hutson, April 9, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. Then A″, B″, C″ are collinear on line X₃₆X₂₃₈ (the trilinear polar of X₈₁). The lines A'A″, B'B″, C'C″ concur in X₂₁. (Randy Hutson, April 9, 2016)

Let O_a be the reflection of the A-excircle in the perpendicular bisector of BC, and define O_b, O_c cyclically. Then X₂₁ is the radical center of O_a, O_b, O_c. (Randy Hutson, April 9, 2016)

Let A_b, A_c, B_c, B_a, C_a, C_b be as in the construction of the Conway circle (see http://mathworld.wolfram.com/ConwayCircle.html). Let O_a be the circumcircle of AAbAc, and define O_b, O_c cyclically. X₂₁ is the radical center of O_a, O_b, O_c; see also X₈ and X₂₇₄. (Randy Hutson, April 9, 2016)

Let J_aJ_bJ_c be the excentral triangle. X₂₁ is the radical center of the circles O(3,4) of triangles J_aBC, J_bCA, J_cAB. (Randy Hutson, July 31 2018)

Lev Emelyanov and Tatiana Emelyanova, A note on the Schiffler point, :cite:`Forum Geometricorum <forum-geometricorum>` 3 (2003) pages 113-116.

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

In the plane of a triangle ABC, let

• A' = A-excenter, and define B' and C' cyclically

• A″ = A'X₃∩BC, and define B″ and C″ cyclically.

The lines AA″, BB″,CC″ concur in X₂₁. (Yuda Chen, April 13, 2022)

X₂₁ is the {X₂,X₃}-harmonic conjugate of X₄₀₄.