X₂₂

EXETER POINT

X₂₂ is the perspector of the circummedial triangle and the tangential triangle; also X₂₂ = X₅₅-of-the-tangential triangle if ABC is acute. See the note just before X₁₆₀₁ for a generalization.

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

X₂₂ (EXETER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Circummedial Triangle △Cm_aCm_bCm_c — vertices Cm_a, Cm_b, Cm_c (Endpoints of medians of △ABC extended to the circumcircle)

• Derived Triangle: Tangential Triangle △Tan_aTan_bTan_c — vertices Tan_a, Tan_b, Tan_c (Bounded by tangents to the circumcircle at A, B, C)

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Concurrence Lines: Cm_a–Tan_a, Cm_b–Tan_b, Cm_c–Tan_c concurring at X₂₂

• Central Lines: {2, 3} (Euler line)

• Related Centers: X₃ (CIRCUMCENTER), X₂ (CENTROID)

Trilinears: \(a(b^{4} + c^{4} - a^{4}) : b(c^{4} + a^{4} - b^{4}) : c(a^{4} + b^{4} - c^{4})\)

Barycentrics: \(a^{2}(b^{4} + c^{4} - a^{4}) : b^{2}(c^{4} + a^{4} - b^{4}) : c^{2}(a^{4} + b^{4} - c^{4})\)

Search Value (6, 9, 13): \(-24.363449636969\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(\sin 2A - \tan \omega : \sin 2B - \tan \omega : :\) (M. Iliev, 5/13/07)

• Barycentrics: \(\tan B + \tan C - \tan A + \tan \omega : :\) (R. Hutson, 10/13/15)

▸ Linear Combinations1 combination

• X₂₂ = \(3 R^{2}\) X₂ - \(S_{\omega}\) X₃

▸ Incident Central Lines659 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}

▸ Functional Relations28 relations

• X₂₂ = reflection of X₃₇₈ in X₃

• X₂₂ = isogonal conjugate of X₆₆

• X₂₂ = isotomic conjugate of X₁₈₀₁₈

• X₂₂ = complement of X₇₃₉₁

• X₂₂ = anticomplement of X₄₂₇

• X₂₂ = circumcircle-inverse of X₈₅₈

• X₂₂ = polar-circle-inverse of X₃₇₉₈₁

• X₂₂ = X₇₆-Ceva conjugate of X₆

• X₂₂ = cevapoint of X₃ and X₁₅₉

• X₂₂ = crosspoint of X₉₉ and X₂₅₀

• X₂₂ = crosssum of X₁₂₅ and X₅₁₂

• X₂₂ = crossdifference of every pair of points on the line X₆₄₇X₈₂₆

• X₂₂ = X_i-beth conjugate of X_j for these (i,j): (643, 345), (833, 22)

• X₂₂ = pole, with respect to circumcircle, of the de Longchamps line

• X₂₂ = isotomic conjugate of the isogonal conjugate of X₂₀₆

• X₂₂ = tangential isogonal conjugate of X₆

• X₂₂ = crosspoint of X₃ and X₁₅₉ wrt both the excentral and tangential triangles

• X₂₂ = homothetic center of the tangential triangle and the orthic triangle of the anticomplementary triangle

• X₂₂ = insimilicenter of circumcircle and tangential circle when ABC is acute

• X₂₂ = exsimilicenter of circumcircle and tangential circle when ABC is obtuse

• X₂₂ = inverse-in-de-Longchamps-circle of X₅₁₈₉

• X₂₂ = inverse-in-{circumcircle, nine-point circle}-inverter of X₂₀₇₂

• X₂₂ = X₇₅-isoconjugate of X₂₃₅₃

• X₂₂ = trilinear pole of line X₂₄₈₅X₈₆₇₃

• X₂₂ = homothetic center of anticomplementary and Ara triangles

• X₂₂ = Thomson-isogonal conjugate of X₅₆₅₄

• X₂₂ = Lucas-isogonal conjugate of X₁₁₄₅₉

• X₂₂ = X₁₉-isoconjugate of X₁₄₃₇₆

Notes

As a point on the Euler line, X₂₂ has Shinagawa coefficients \((E + 2F, -2E - 2F)\).

Let L_a be the polar of X₃ wrt the A-power circle, and define L_b, L_c cyclically. Let A' = L_b∩L_c, B' = L_c∩L_a, C' = L_a∩L_b. The triangle A'B'C' is homothetic to the anticomplementary triangle, and the center of homothety is X₂₂. (Randy Hutson, September 5, 2015)

X₂₂ is the {X₃,X₂₅}-harmonic conjugate of X₂.