X(21) = SCHIFFLER POINT

Trilinears

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

Barycentrics

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

Notes

As a point on the Euler line, X(21) has Shinagawa coefficients ($aSA$, abc - $aSA$).

The name of this point honors Kurt Schiffler.

Let A’B’C’ be the incentral triangle of ABC, and let LA be the reflection of line B’C’ in line BC; define LB and LC cyclically. The triangle formed by the lines LA, LB, LC is perspective to ABC, and the perspector is X(21). (Randy Hutson, 9/23/2011)

Write I for the incenter; the Euler lines of the four triangles IBC, ICA, IAB, and ABC concur in X(21). This configuration extends to Kirikami-Schiffler points and generalizations found by Peter Moses, as introduced just before X(3648).

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 X(21). (Randy Hutson, April 9, 2016)

Let A’B’C’ be the Feuerbach triangle. Let La be the tangent to the nine-point circle at A’, and define Lb and Lc cyclically. Let A″ be the isogonal conjugate of the trilinear pole of La, and define B″ and C″ cyclically. Let A*B*C* be the circumcevian triangle, wrt A″B″C″, of X(1). The lines AA*, BB*, CC* concur in X(21). (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 X(36)X(238) (the trilinear polar of X(81)). The lines A’A″, B’B″, C’C″ concur in X(21). (Randy Hutson, April 9, 2016)

Let Oa be the reflection of the A-excircle in the perpendicular bisector of BC, and define Ob, Oc cyclically. Then X(21) is the radical center of Oa, Ob, Oc. (Randy Hutson, April 9, 2016)

Let Ab, Ac, Bc, Ba, Ca, Cb be as in the construction of the Conway circle (see http://mathworld.wolfram.com/ConwayCircle.html). Let Oa be the circumcircle of AAbAc, and define Ob, Oc cyclically. X(21) is the radical center of Oa, Ob, Oc; see also X(8) and X(274). (Randy Hutson, April 9, 2016)

Let A’B’C’ be the excentral triangle. X(21) is the radical center of the circles O(3,4) of triangles A’BC, B’CA, C’AB. (Randy Hutson, July 31 2018)

If you have The Geometer’s Sketchpad, you can view Schiffler point. If you have GeoGebra, you can view Schiffler point.

Lev Emelyanov and Tatiana Emelyanova, A note on the Schiffler point, 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’X(3). (Yuda Chen, April 13, 2022)

X(21) lies on these 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}

X(21) is the {X(2),:ref:X(3) <X(3)>}-harmonic conjugate of X(404). For a list of other harmonic conjugates of X(21), click Tables at the top of this page.

X(21) = midpoint of X(1) and X(191)

X(21) = reflection of X(3651) in X(3)

X(21) = isogonal conjugate of X(65)

X(21) = isotomic conjugate of X(1441)

X(21) = circumcircle-inverse of X(1325)

X(21) = polar-circle-inverse of X(37982)

X(21) = orthoptic-circle-of-Steiner-inellipse-inverse of complement of X(37959)

X(21) = orthoptic-circle-of-Steiner-circumellipse-inverse of anticomplement of X(37959)

X(21) = complement of X(2475)

X(21) = anticomplement of X(442)

X(21) = X(i)-Ceva conjugate of X(j) for these (i,j): (86,81), (261,333)

X(21) = cevapoint of X(i) and X(j) for these (i,j): (1,3), (9,55), (1805,1806)

X(21) = 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)

X(21) = crosspoint of X(i) and X(j) for these {i,j}: {86,333}, {1805,1806}

X(21) = crosssum of X(i) and X(j) for these (i,j): (1,1046), (42,1400), (1254,1425), (1402,1409)

X(21) = crossdifference of every pair of points on line X(647)X(661)

X(21) = X(i)-Hirst inverse of X(j) for these (i,j): (2,448), (3,416), (4,425)

X(21) = X(i)-beth conjugate of X(j) for these (i,j): (21,58), (99,21), (643,21), (1043,1043), (1098,21)

X(21) = intersection of tangents at X(1) and X(3) to the Stammler hyperbola

X(21) = X(54)-of-2nd-circumperp-triangle

X(21) = X(3574)-of-excentral-triangle

X(21) = crosspoint of X(1) and X(3) wrt the excentral triangle

X(21) = crosspoint of X(1) and X(3) wrt the tangential triangle

X(21) = trilinear pole of line X(521)X(650)

X(21) = similitude center of ABC and X(1)-Brocard triangle

X(21) = X(i)-isoconjugate of X(j) for these (i,j): (6,226), (75,1402)

X(21) = {X(1),:ref:X(63) <X(63)>}-harmonic conjugate of X(3868)

X(21) = perspector of 2nd circumperp triangle and cross-triangle of ABC and 2nd circumperp triangle

X(21) = perspector of ABC and cross-triangle of ABC and 1st Conway triangle

X(21) = perspector of Gemini triangles 1 and 8

X(21) = barycentric product of Feuerbach hyperbola intercepts of line X(2)X(6)