WHAT!IS THE EXITER POINT COPYPSTA! IS IT THAT THING ABOUT the trilinaer polar wtv

yeah whatever that means

sus

should it be 3rd

i did it like that on purpose

X(22) = EXETER POINT
Trilinears a(b4 + c4 - a4) : b(c4 + a4 - a4) : c(a4 + b4 - c4)
Barycentrics a2(b4 + c4 - a4) : b2(c4 + a4 - a4) : c2(a4 + b4 - c4)
The perspector of the circummedial triangle and the tangential triangle; also X(22) = X(55)-of-the-tangential triangle if ABC is acute.

X(22) lies on these lines:
2,3 6,251 35,612 36,614 51,182 56,977 69,159 98,925 99,305 100,197 110,154 157,183 160,325 161,343 184,511 232,577

X(22) = reflection of X(378) about X(3)
X(22) = isogonal conjugate of X(66)
X(22) = inverse of X(858) in the circumcircle
X(22) = anticomplement of X(427)
X(22) = X(76)-Ceva conjugate of X(6)
X(22) = cevapoint of X(3) and X(159)
X(22) = crosspoint of X(99) and X(250)
X(22) = X(I)-beth conjugate of X(J) for these (I,J): (643,345), (833,22)

Let ABC be any given triangle. Let the medians through the vertices A, B, C meet the circumcircle of ABC at A', B', C' respectively. Let DEF be the triangle formed by the tangents at A, B, C to the circumcircle of ABC. (Let D be the vertex opposite to the side formed by the tangent at the vertex A, E be the vertex opposite to the side formed by the tangent at the vertex B, and F be the vertex opposite to the side formed by the tangent at the vertex C.) The lines through DA', EB', FC' are concurrent. The point of concurrence is the Exeter point of ABC.

Trilinears a(b4 + c4 - a4) : b(c4 + a4 - b4) : c(a4 + b4 - c4)
Barycentrics a2(b4 + c4 - a4) : b2(c4 + a4 - b4) : c2(a4 + b4 - c4)
Barycentrics sin 2A - tan ω : sin 2B - tan ω : : (M. Iliev, 5/13/07)
Barycentrics tan B + tan C - tan A + tan ω : : (R. Hutson, 10/13/15)
X(22) = 3 R^2 X(2) - SW X(3)
As a point on the Euler line, X(22) has Shinagawa coefficients (E + 2F, -2E - 2F).

X(22) is the perspector of the circummedial triangle and the tangential triangle; also X(22) = X(55)-of-the-tangential triangle if ABC is acute. See the note just before X(1601) for a generalization.

Let La be the polar of X(3) wrt the A-power circle, and define Lb, Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. The triangle A'B'C' is homothetic to the anticomplementary triangle, and the center of homothety is X(22). (Randy Hutson, September 5, 2015)

For a generalization and related references, see Peter Csiba and László Nément, Mathematics 2021: "Some Properties of the Exeter Transformation". The Exeter transformation is closely related to TCC Perspectors, introducted in 2003 in the preamble just before X(1601). This subject is developed in I. Minevich and P. Morton, International Journal of Geometry 2017, "Synthetic foundations of cevian geometry, IV"

If you have The Geometer's Sketchpad, you can view Exeter point.

If you have GeoGebra, you can view Exeter point.

X(22) lies on these lines: {1, 5310}, {2, 3}, {6, 251}, {8, 8193}, {9, 5314}, {10, 8185}, {11, 9673}, {12, 9658}, {31, 5329}, {32, 1194}, {35, 612}, {36, 614}, {40, 9626}, {42, 37576}, {49, 16266}, {51, 182}, {52, 7592}, {54, 36747}, {55, 3100}, {56, 977}, {57, 7293}, {63, 3220}, {64, 11440}, {66, 34177}, {68, 32048}, {69, 159}, {74, 2931}, {75, 21407}, {76, 1799}, {81, 36740}, {83, 41928}, {97, 19189}, {98, 925}, {99, 305}, {100, 197}, {105, 13397}, {107, 15466}, {110, 154}, {111, 2079}, {112, 3162}, {114, 23217}, {125, 41674}, {127, 11605}, {132, 35969}, {141, 20987}, {143, 36753}, {145, 8192}, {146, 9919}, {147, 9861}, {148, 13175}, {149, 13222}, {153, 9913}, {155, 1614}, {156, 6101}, {157, 183}, {160, 325}, {161, 343}, {165, 9590}, {184, 511}, {187, 1196}, {193, 19119}, {194, 9917}, {195, 12226}, {198, 27396}, {206, 3313}, {216, 10311}, {220, 26911}, {221, 19367}, {230, 8553}, {232, 577}, {238, 27661}, {262, 40393}, {264, 1629}, {280, 7172}, {315, 23208}, {316, 14558}, {321, 23847}, {323, 3167}, {324, 33971}, {347, 1617}, {348, 39732}, {353, 11173}, {371, 9683}, {373, 17508}, {385, 3164}, {386, 9571}, {388, 10831}, {389, 10984}, {390, 16541}, {399, 12219}, {476, 2697}, {477, 16167}, {485, 35776}, {486, 35777}, {487, 9921}, {488, 9922}, {491, 26307}, {492, 26306}, {497, 10832}, {515, 15177}, {519, 37546}, {524, 35707}, {543, 3455}, {567, 39522}, {569, 5446}, {573, 9570}, {574, 9699}, {575, 15004}, {576, 13366}, {595, 2922}, {599, 19596}, {616, 9916}, {617, 9915}, {627, 22657}, {628, 22656}, {638, 8996}, {669, 6563}, {675, 1305}, {689, 40362}, {842, 10420}, {901, 10016}, {907, 40189}, {930, 15959}, {940, 4265}, {956, 33090}, {958, 9712}, {962, 9911}, {991, 1790}, {999, 17024}, {1001, 20988}, {1007, 44180}, {1030, 5275}, {1040, 24611}, {1069, 9638}, {1078, 40022}, {1092, 10282}, {1112, 19154}, {1147, 9707}, {1151, 9694}, {1152, 12224}, {1154, 18445}, {1181, 5889}, {1184, 1627}, {1192, 43601}, {1199, 37493}, {1216, 10539}, {1225, 2934}, {1269, 23365}, {1270, 5594}, {1271, 5595}, {1289, 39436}, {1294, 1302}, {1295, 9058}, {1296, 14657}, {1311, 41906}, {1324, 7081}, {1351, 1994}, {1352, 31383}, {1369, 11641}, {1376, 9713}, {1383, 5024}, {1384, 5354}, {1407, 26910}, {1437, 37482}, {1460, 17126}, {1473, 3218}, {1486, 1621}, {1495, 3098}, {1498, 2917}, {1602, 1626}, {1603, 2933}, {1605, 2925}, {1606, 2926}, {1609, 7735}, {1611, 5023}, {1612, 7742}, {1613, 2076}, {1615, 2919}, {1616, 2920}, {1620, 2929}, {1634, 6148}, {1637, 25644}, {1661, 38918}, {1670, 8881}, {1671, 8880}, {1691, 3981}, {1714, 5358}, {1760, 4123}, {1843, 9813}, {1853, 23293}, {1899, 3580}, {1915, 3094}, {1972, 40870}, {1974, 11574}, {1975, 8024}, {1992, 32621}, {2000, 21370}, {2056, 5104}, {2077, 36984}, {2172, 4456}, {2178, 26242}, {2192, 11446}, {2194, 4259}, {2370, 9059}, {2393, 27365}, {2493, 38872}, {2693, 9060}, {2770, 14729}, {2777, 22109}, {2782, 5986}, {2799, 42659}, {2888, 9920}, {2896, 9918}, {2923, 24303}, {2924, 24304}, {2930, 14682}, {2932, 35221}, {2967, 23606}, {2975, 22654}, {3006, 23361}, {3007, 18613}, {3011, 36152}, {3051, 5017}, {3052, 5078}, {3066, 11451}, {3085, 10037}, {3086, 10046}, {3124, 38880}, {3197, 11445}, {3219, 7085}, {3291, 5206}, {3292, 44110}, {3295, 9538}, {3410, 18440}, {3434, 10829}, {3436, 10830}, {3447, 5968}, {3448, 12310}, {3504, 8782}, {3556, 3869}, {3563, 13398}, {3567, 36752}, {3576, 9625}, {3592, 34516}, {3594, 34515}, {3616, 11365}, {3648, 16119}, {3681, 12329}, {3705, 15654}, {3721, 21771}, {3734, 8891}, {3736, 44118}, {3746, 9643}, {3757, 23850}, {3781, 26885}, {3784, 26884}, {3819, 5651}, {3868, 37547}, {3870, 40910}, {3871, 20020}, {3873, 22769}, {3926, 40123}, {3955, 26892}, {3964, 37668}, {4057, 20294}, {4252, 33774}, {4260, 5320}, {4383, 5096}, {4440, 24822}, {4549, 32111}, {4550, 16194}, {5010, 5268}, {5013, 9608}, {5050, 9777}, {5085, 5640}, {5092, 5943}, {5093, 16981}, {5134, 24054}, {5138, 40952}, {5157, 9969}, {5172, 29665}, {5188, 42671}, {5191, 38553}, {5201, 14614}, {5204, 7292}, {5217, 5297}, {5272, 7280}, {5276, 36744}, {5304, 8573}, {5324, 24597}, {5406, 12305}, {5407, 12306}, {5408, 11825}, {5409, 8989}, {5412, 11514}, {5413, 11513}, {5462, 13336}, {5480, 37649}, {5523, 13854}, {5552, 26309}, {5562, 6759}, {5601, 8190}, {5602, 8191}, {5621, 9140}, {5687, 33091}, {5695, 23848}, {5706, 41723}, {5858, 14179}, {5859, 14173}, {5864, 11126}, {5865, 11127}, {5866, 19583}, {5890, 37489}, {5897, 9064}, {5907, 26883}, {5921, 39879}, {5938, 6031}, {5966, 14656}, {5976, 14713}, {5987, 13188}, {6090, 8780}, {6193, 9908}, {6194, 22655}, {6198, 9645}, {6200, 8854}, {6221, 9695}, {6223, 9910}, {6224, 9912}, {6225, 9914}, {6241, 8718}, {6243, 12161}, {6284, 9672}, {6337, 40125}, {6360, 20999}, {6396, 8855}, {6462, 8194}, {6463, 8195}, {6467, 40318}, {6480, 32567}, {6481, 32574}, {6503, 7710}, {6515, 6776}, {6527, 15589}, {6560, 18289}, {6561, 18290}, {6688, 22112}, {6781, 9745}, {7071, 9539}, {7083, 17127}, {7193, 26893}, {7262, 24436}, {7354, 9659}, {7585, 19006}, {7586, 19005}, {7669, 8667}, {7689, 10575}, {7750, 15270}, {7754, 8267}, {7761, 21248}, {7774, 20775}, {7779, 20794}, {7781, 19568}, {7787, 10790}, {7802, 16275}, {7823, 8878}, {7842, 30747}, {7878, 42037}, {7893, 19597}, {7910, 30785}, {7998, 17811}, {7999, 43598}, {8053, 20291}, {8125, 8131}, {8126, 8132}, {8276, 9540}, {8277, 13935}, {8280, 35820}, {8281, 35821}, {8546, 8584}, {8588, 20481}, {8591, 9876}, {8680, 24321}, {8717, 14855}, {8743, 10316}, {8793, 19613}, {8879, 41361}, {8903, 8904}, {8911, 26875}, {8939, 19406}, {8943, 19407}, {8972, 13889}, {9056, 41904}, {9057, 41905}, {9070, 39435}, {9084, 20187}, {9123, 34519}, {9209, 39228}, {9536, 11406}, {9537, 10306}, {9732, 10132}, {9733, 10133}, {9744, 23195}, {9781, 43651}, {9786, 10574}, {9833, 14516}, {9865, 23173}, {9874, 12411}, {9924, 12272}, {9927, 11750}, {9934, 12825}, {9937, 11411}, {9967, 44077}, {10192, 11064}, {10203, 13423}, {10263, 32046}, {10314, 10979}, {10330, 25046}, {10519, 14826}, {10527, 26308}, {10528, 10834}, {10529, 10835}, {10540, 15068}, {10541, 12834}, {10545, 31860}, {10546, 41424}, {10602, 37784}, {10605, 15072}, {10606, 11454}, {10641, 11516}, {10642, 11515}, {10733, 19457}, {10982, 13434}, {11012, 36986}, {11061, 32262}, {11174, 41328}, {11202, 36987}, {11245, 37644}, {11363, 37613}, {11422, 11477}, {11424, 13598}, {11433, 25406}, {11439, 15811}, {11443, 17813}, {11444, 17814}, {11447, 17819}, {11448, 17820}, {11449, 17821}, {11452, 17826}, {11453, 17827}, {11455, 11472}, {11456, 13754}, {11457, 12359}, {11459, 14157}, {11464, 37483}, {11480, 37776}, {11481, 37775}, {11511, 44102}, {11550, 21243}, {11580, 15655}, {11610, 22075}, {11629, 14184}, {11630, 14183}, {11643, 33998}, {11671, 14652}, {11820, 35450}, {11898, 14683}, {12017, 15018}, {12118, 19908}, {12160, 19347}, {12164, 43605}, {12203, 40814}, {12221, 12978}, {12222, 12979}, {12256, 12972}, {12257, 12973}, {12270, 17835}, {12271, 17836}, {12273, 17838}, {12274, 17839}, {12275, 17842}, {12276, 17840}, {12277, 17843}, {12278, 17845}, {12280, 17846}, {12284, 15085}, {12289, 12293}, {12383, 12412}, {12384, 12413}, {12414, 12849}, {12429, 34799}, {12824, 15462}, {12827, 36201}, {12893, 16111}, {13009, 13055}, {13010, 13056}, {13015, 17841}, {13016, 17844}, {13289, 16163}, {13321, 15037}, {13330, 14153}, {13340, 22115}, {13346, 13367}, {13348, 43652}, {13352, 18475}, {13394, 23292}, {13421, 32136}, {13567, 18911}, {13630, 37490}, {13638, 44192}, {13678, 13680}, {13758, 44193}, {13798, 13800}, {13858, 36329}, {13859, 35751}, {13941, 13943}, {14370, 17042}, {14389, 31670}, {14547, 22390}, {14577, 15355}, {14602, 43183}, {14673, 34186}, {14793, 24239}, {14852, 25739}, {14927, 32064}, {15024, 15805}, {15033, 37506}, {15043, 37514}, {15053, 20791}, {15060, 33533}, {15069, 15581}, {15109, 31489}, {15241, 31842}, {15302, 15815}, {15360, 43273}, {15512, 33495}, {15513, 40350}, {15520, 44111}, {15578, 23332}, {15588, 35213}, {15801, 19468}, {15812, 26156}, {15931, 30265}, {16030, 43768}, {16102, 39346}, {16261, 32620}, {16318, 42459}, {16472, 31757}, {16681, 23339}, {16989, 40981}, {16990, 22062}, {16998, 18666}, {17018, 37580}, {17093, 38859}, {17150, 20247}, {17165, 20249}, {17824, 32338}, {17837, 22534}, {17907, 41375}, {18124, 34436}, {18287, 39653}, {18392, 18405}, {18436, 32139}, {18438, 34397}, {18616, 20911}, {18617, 33936}, {18912, 41587}, {19122, 19132}, {19131, 39588}, {19137, 44091}, {19153, 22151}, {19167, 19180}, {19357, 34148}, {19412, 19430}, {19413, 19431}, {19588, 20080}, {19724, 19759}, {19725, 19760}, {19785, 41230}, {19798, 19841}, {19799, 19842}, {19835, 19845}, {20045, 20222}, {20127, 32227}, {20676, 28395}, {20878, 23385}, {20998, 21001}, {21072, 29065}, {21167, 35283}, {21368, 24430}, {22089, 30474}, {22135, 34137}, {22241, 32817}, {22647, 22658}, {22676, 35278}, {22802, 23358}, {23061, 37672}, {23115, 39575}, {23128, 41480}, {23216, 36849}, {23368, 23374}, {23380, 26232}, {23381, 32929}, {23383, 26230}, {23843, 26227}, {23864, 26248}, {23958, 26866}, {24163, 30117}, {24686, 25343}, {25335, 34437}, {25524, 29666}, {26228, 37579}, {26275, 39478}, {26302, 26394}, {26303, 26418}, {26304, 26494}, {26305, 26503}, {26895, 26909}, {26912, 26953}, {26913, 26958}, {29680, 37564}, {30270, 36212}, {30435, 34482}, {32248, 32276}, {32354, 32357}, {32379, 41590}, {32458, 39644}, {32762, 38227}, {32911, 36741}, {33854, 36743}, {33974, 37667}, {34013, 36521}, {34247, 36559}, {34424, 36836}, {34425, 36843}, {34565, 39561}, {34809, 37689}, {34966, 41615}, {35260, 37669}, {35325, 38663}, {36988, 37813}, {37492, 37685}, {37511, 44080}, {37516, 44085}, {37517, 44109}, {37779, 39899}, {38738, 39854}, {38749, 39825}, {39172, 40358}, {39807, 39820}, {39836, 39849}, {40120, 44064}, {40643, 41262}, {41447, 41468}, {41448, 41469}, {41594, 41730}, {41602, 41736}, {41605, 41740}, {41612, 41743}, {43816, 43829}

X(22) is the {X(3),X(25)}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(22), click Tables at the top of this page.

X(22) = reflection of X(378) in X(3)
X(22) = isogonal conjugate of X(66)
X(22) = isotomic conjugate of X(18018)
X(22) = complement of X(7391)
X(22) = anticomplement of X(427)
X(22) = circumcircle-inverse of X(858)
X(22) = polar-circle-inverse of X(37981)
X(22) = X(76)-Ceva conjugate of X(6)
X(22) = cevapoint of X(3) and X(159)
X(22) = crosspoint of X(99) and X(250)
X(22) = crosssum of X(125) and X(512)
X(22) = crossdifference of every pair of points on the line X(647)X(826)
X(22) = X(i)-beth conjugate of X(j) for these (i,j): (643,345), (833,22)
X(22) = pole, with respect to circumcircle, of the de Longchamps line
X(22) = isotomic conjugate of the isogonal conjugate of X(206)
X(22) = tangential isogonal conjugate of X(6)
X(22) = crosspoint of X(3) and X(159) wrt both the excentral and tangential triangles
X(22) = homothetic center of the tangential triangle and the orthic triangle of the anticomplementary triangle
X(22) = insimilicenter of circumcircle and tangential circle when ABC is acute
X(22) = exsimilicenter of circumcircle and tangential circle when ABC is obtuse
X(22) = inverse-in-de-Longchamps-circle of X(5189)
X(22) = inverse-in-{circumcircle, nine-point circle}-inverter of X(2072)
X(22) = X(75)-isoconjugate of X(2353)
X(22) = trilinear pole of line X(2485)X(8673)
X(22) = homothetic center of anticomplementary and Ara triangles
X(22) = Thomson-isogonal conjugate of X(5654)
X(22) = Lucas-isogonal conjugate of X(11459)
X(22) = X(19)-isoconjugate of X(14376)