X₅

NINE-POINT CENTER

X₅ is the center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X₄, B-to-X₄, C-to-X₄. The radius of the nine-point circle is one-half the circumradius.

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

X₅ (NINE-POINT CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Auxiliary Points: M_a, M_b, M_c — Side midpoints of BC, CA, AB

• Auxiliary Points: H_a, H_b, H_c — Altitude feet on BC, CA, AB

• Circle: Nine-Point Circle (nine_pointcircle) — Passes through side midpoints M_a, M_b, M_c and altitude feet H_a, H_b, H_c; centered at X₅

• Central Lines: {3, 4} (Euler line)

• Related Centers: X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER)

Trilinears: \(\cos(B - C) : \cos(C - A) : \cos(A - B)\)

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

Tripolars: \(\sqrt{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}-3 a^{4} c^{2}+3 a^{2} b^{2} c^{2}+b^{4} c^{2}+3 a^{2} c^{4}+b^{2} c^{4}-c^{6}} : :\)

Search Value (6, 9, 13): \(0.552871741635\)

▸ Alternate Coordinates11 forms

• Trilinears: \(\cos A + 2 \cos B \cos C : :\)

• Trilinears: \(\cos A - 2 \sin B \sin C\)

• Trilinears: \(bc[a^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}] : :\)

• Trilinears: \(2 \sec A + \sec B \sec C : :\)

• Trilinears: \(\cos B \cos C + \sin B \sin C : :\)

• Trilinears: \(\cos^{2}(B/2 - C/2) - \sin^{2}(B/2 - C/2) : :\)

• Trilinears: \(\cos A' : :\), where A' is the angle formed by internal tangents to incircle and A-excircle

• Trilinears: \(\cos A'' : :\), where A″ is the angle formed by external tangents to B- and C-excircles

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

• Barycentrics: \(S^{2} + S_{B} S_{C} : :\)

• Barycentrics: \(1 + \cot B \cot C : :\)

▸ Linear Combinations2 combinations

• X₅ = \(3\) X₂ + X₄ = \(3\) X₂ - X₃ = X₃ + X₄

• X₅ = \((2 + J)\) X₁₁₁₃ + \((2 - J)\) X₁₁₁₄

X₅ lies on the Napoleon cubic (also known as the Feuerbach cubic).

Related Collection 1 Topic Guides

T_3: Complement & Anticomplement • T_4: Harmonic Conjugate & Cross-Ratio • T_6: The Euler Line • T_9: The Feuerbach Line • T_15: The Circumcircle • T_16: The Nine-Point (Feuerbach) Circle • T_19: The Brocard & Orthocentroidal Circles • T_20: Medial & Anticomplementary Triangles • T_21: Orthic & Tangential Triangles

▸ Incident Central Lines718 lines

{1, 11}, {2, 3}, {6, 68}, {7, 5704}, {8, 1389}, {9, 1729}, {10, 517}, {13, 18}, {14, 17}, {15, 2913}, {16, 2912}, {19, 8141}, {32, 230}, {33, 1062}, {34, 1060}, {35, 3583}, {36, 3585}, {39, 114}, {40, 1698}, {46, 1836}, {47, 5348}, {49, 54}, {51, 52}, {53, 216}, {55, 498}, {56, 499}, {57, 1728}, {60, 5397}, {64, 4846}, {65, 1737}, {67, 9970}, {69, 1351}, {72, 908}, {74, 3521}, {76, 262}, {78, 3419}, {79, 1749}, {83, 98}, {84, 5437}, {85, 1565}, {93, 6344}, {94, 9221}, {96, 1166}, {97, 4994}, {99, 5966}, {100, 10738}, {101, 10739}, {102, 10740}, {103, 10741}, {104, 5253}, {105, 10743}, {106, 10744}, {107, 10745}, {108, 10746}, {109, 10747}, {111, 10748}, {112, 10749}, {113, 125}, {116, 118}, {117, 124}, {120, 5511}, {121, 2885}, {122, 133}, {126, 5512}, {127, 132}, {128, 137}, {129, 130}, {131, 136}, {141, 211}, {142, 971}, {145, 10247}, {146, 10620}, {147, 3329}, {148, 7783}, {149, 3871}, {153, 12773}, {154, 9833}, {156, 184}, {165, 7965}, {169, 6506}, {171, 3073}, {181, 10407}, {182, 206}, {183, 315}, {187, 3054}, {191, 5535}, {193, 5093}, {194, 7777}, {195, 1994}, {214, 6246}, {217, 1625}, {222, 8757}, {225, 1465}, {226, 912}, {236, 8130}, {238, 3072}, {252, 1157}, {264, 1093}, {273, 6356}, {275, 2055}, {276, 6528}, {298, 634}, {299, 633}, {302, 622}, {303, 621}, {311, 1225}, {312, 3695}, {316, 1078}, {318, 2968}, {324, 6663}, {329, 2095}, {339, 1235}, {354, 13407}, {356, 3608}, {371, 590}, {372, 615}, {385, 7762}, {386, 1834}, {388, 999}, {389, 5448}, {390, 7678}, {392, 1512}, {394, 10982}, {399, 3448}, {484, 5445}, {487, 12313}, {488, 12314}, {491, 637}, {492, 638}, {493, 8220}, {494, 8221}, {497, 3085}, {515, 1125}, {516, 3579}, {518, 10916}, {519, 3813}, {523, 6757}, {524, 576}, {528, 8715}, {529, 8666}, {538, 7764}, {539, 1493}, {542, 575}, {543, 9771}, {551, 5882}, {566, 9220}, {568, 3567}, {570, 1879}, {572, 2126}, {573, 1213}, {574, 3055}, {577, 6748}, {578, 1147}, {579, 1901}, {580, 5127}, {581, 5453}, {582, 1754}, {583, 8818}, {598, 7607}, {599, 11477}, {601, 750}, {602, 748}, {611, 12589}, {613, 12588}, {616, 13103}, {617, 13102}, {618, 629}, {619, 630}, {620, 6721}, {641, 6250}, {642, 6251}, {671, 7608}, {698, 8149}, {754, 7780}, {758, 5694}, {804, 11615}, {842, 1287}, {920, 1454}, {925, 2383}, {930, 6592}, {938, 3487}, {944, 3616}, {950, 13411}, {956, 3436}, {958, 10526}, {962, 5657}, {986, 3944}, {993, 4999}, {997, 5794}, {1001, 10198}, {1007, 3926}, {1056, 5261}, {1058, 5274}, {1069, 10055}, {1071, 5249}, {1073, 1217}, {1087, 2599}, {1089, 3703}, {1090, 1091}, {1092, 5651}, {1111, 3665}, {1112, 12358}, {1117, 3470}, {1131, 3317}, {1132, 3316}, {1139, 3393}, {1140, 3370}, {1145, 7704}, {1151, 5418}, {1152, 5420}, {1155, 1770}, {1158, 5880}, {1160, 5590}, {1161, 5591}, {1173, 1487}, {1181, 1899}, {1199, 3410}, {1211, 5752}, {1212, 5179}, {1214, 1838}, {1249, 8888}, {1270, 11917}, {1271, 11916}, {1297, 12918}, {1327, 6426}, {1328, 6425}, {1350, 3763}, {1376, 10525}, {1393, 7069}, {1420, 9613}, {1441, 3007}, {1447, 4911}, {1490, 5787}, {1495, 11572}, {1498, 1853}, {1499, 11182}, {1511, 5972}, {1519, 3753}, {1537, 7705}, {1538, 8582}, {1539, 2777}, {1587, 3069}, {1588, 3068}, {1601, 3432}, {1614, 5012}, {1621, 11491}, {1697, 9614}, {1706, 12700}, {1709, 12679}, {1714, 4383}, {1724, 5398}, {1750, 8726}, {1768, 7701}, {1788, 4295}, {1843, 9967}, {1848, 1871}, {1861, 1872}, {1916, 3399}, {1935, 3075}, {1936, 3074}, {1975, 6390}, {1986, 7723}, {1990, 5158}, {1992, 11482}, {2052, 13599}, {2066, 9646}, {2067, 9661}, {2077, 11826}, {2086, 9490}, {2098, 12647}, {2099, 10573}, {2120, 2121}, {2241, 9665}, {2242, 9650}, {2482, 9880}, {2486, 2783}, {2549, 5013}, {2550, 9709}, {2551, 9708}, {2595, 3460}, {2601, 2602}, {2607, 2957}, {2635, 4303}, {2646, 10572}, {2682, 10568}, {2771, 5883}, {2781, 6698}, {2792, 4672}, {2794, 6036}, {2797, 8552}, {2800, 3754}, {2801, 12005}, {2802, 10284}, {2826, 3837}, {2829, 5450}, {2883, 5892}, {2887, 3831}, {2896, 7616}, {2963, 2965}, {2971, 2974}, {2975, 5080}, {2979, 7999}, {3006, 3701}, {3035, 5840}, {3053, 7737}, {3057, 10039}, {3058, 3584}, {3060, 6243}, {3096, 7934}, {3098, 7914}, {3120, 5492}, {3157, 10071}, {3167, 6193}, {3258, 11749}, {3272, 3609}, {3284, 6749}, {3303, 10056}, {3304, 10072}, {3306, 13226}, {3314, 7912}, {3333, 5290}, {3338, 10404}, {3357, 5893}, {3359, 12705}, {3368, 5401}, {3381, 3382}, {3406, 3407}, {3434, 5552}, {3462, 3463}, {3468, 3469}, {3488, 5703}, {3576, 3624}, {3581, 10545}, {3582, 5270}, {3586, 3601}, {3617, 8148}, {3618, 5050}, {3619, 10519}, {3622, 7967}, {3626, 11278}, {3629, 5097}, {3630, 7882}, {3631, 7896}, {3636, 13607}, {3649, 5693}, {3654, 7991}, {3656, 3679}, {3670, 3782}, {3673, 7179}, {3687, 5295}, {3705, 4385}, {3734, 3788}, {3737, 8819}, {3739, 12490}, {3742, 12675}, {3785, 9752}, {3812, 3838}, {3819, 5447}, {3823, 12393}, {3833, 6701}, {3867, 11574}, {3874, 6583}, {3880, 10915}, {3911, 4292}, {3913, 11235}, {3917, 10625}, {3931, 5530}, {3947, 5045}, {3972, 7857}, {4004, 10273}, {4030, 4894}, {4045, 6683}, {4293, 5229}, {4294, 5218}, {4297, 10165}, {4299, 5204}, {4302, 5217}, {4311, 5126}, {4323, 11041}, {4354, 9629}, {4413, 10310}, {4417, 10449}, {4420, 5178}, {4425, 9959}, {4511, 5086}, {4550, 7689}, {4662, 12612}, {4668, 11224}, {4855, 9945}, {4861, 5176}, {4885, 8760}, {5007, 5306}, {5010, 5326}, {5015, 7081}, {5024, 7738}, {5038, 11646}, {5041, 5355}, {5082, 7080}, {5092, 6704}, {5099, 13162}, {5119, 12701}, {5131, 5442}, {5171, 7761}, {5181, 8263}, {5188, 6249}, {5221, 11544}, {5223, 6067}, {5224, 10446}, {5233, 9534}, {5237, 5350}, {5238, 5349}, {5248, 5842}, {5251, 11012}, {5257, 10445}, {5259, 6253}, {5286, 7736}, {5309, 7772}, {5334, 11485}, {5335, 11486}, {5339, 10654}, {5340, 10653}, {5354, 10339}, {5395, 7612}, {5412, 10897}, {5413, 10898}, {5422, 7592}, {5471, 6783}, {5472, 6782}, {5544, 5656}, {5550, 5731}, {5597, 8200}, {5598, 8207}, {5599, 8196}, {5600, 8203}, {5601, 11875}, {5602, 11876}, {5643, 5655}, {5658, 9799}, {5745, 12572}, {5814, 11679}, {5890, 11451}, {5895, 10606}, {5908, 5909}, {5913, 6032}, {5925, 8567}, {5944, 6689}, {5947, 5948}, {5950, 5952}, {5961, 8146}, {5962, 13557}, {6043, 11992}, {6054, 7827}, {6055, 10991}, {6118, 8180}, {6119, 8184}, {6130, 9517}, {6150, 10615}, {6152, 12606}, {6153, 11692}, {6174, 10993}, {6179, 7812}, {6191, 7345}, {6192, 7344}, {6194, 7938}, {6221, 6459}, {6223, 12684}, {6224, 12747}, {6225, 13093}, {6237, 11435}, {6238, 11436}, {6241, 10574}, {6256, 10200}, {6291, 12603}, {6329, 12007}, {6361, 9812}, {6398, 6460}, {6406, 12604}, {6407, 9692}, {6417, 7582}, {6418, 7581}, {6419, 8960}, {6433, 12819}, {6434, 12818}, {6449, 9541}, {6462, 11949}, {6463, 11950}, {6515, 9777}, {6523, 10002}, {6599, 12660}, {6662, 13409}, {6669, 6694}, {6670, 6695}, {6671, 6673}, {6672, 6674}, {6692, 6705}, {6703, 13323}, {6735, 10914}, {6736, 13600}, {6769, 8580}, {7013, 10400}, {7028, 8129}, {7160, 12856}, {7198, 7272}, {7280, 7294}, {7596, 8228}, {7615, 11184}, {7620, 11165}, {7691, 11016}, {7693, 12307}, {7694, 9756}, {7703, 11439}, {7709, 7864}, {7743, 9957}, {7754, 7774}, {7758, 9766}, {7760, 7858}, {7766, 7921}, {7768, 7809}, {7771, 7802}, {7778, 7795}, {7779, 7941}, {7784, 7800}, {7786, 7790}, {7787, 7806}, {7793, 7823}, {7794, 7821}, {7801, 7888}, {7803, 7851}, {7810, 7873}, {7811, 7860}, {7818, 7854}, {7820, 7874}, {7822, 7867}, {7826, 7845}, {7830, 7842}, {7831, 7911}, {7832, 7899}, {7835, 7940}, {7836, 7925}, {7846, 7942}, {7852, 7889}, {7855, 7903}, {7856, 7878}, {7859, 7919}, {7875, 7932}, {7877, 7926}, {7893, 7900}, {7898, 7904}, {7935, 8722}, {7998, 13340}, {8014, 11555}, {8015, 11556}, {8069, 10320}, {8085, 8087}, {8086, 8088}, {8121, 8123}, {8122, 8124}, {8158, 8165}, {8212, 8222}, {8213, 8223}, {8280, 8855}, {8281, 8854}, {8351, 8379}, {8377, 8380}, {8378, 8381}, {8538, 8541}, {8591, 12355}, {8725, 9751}, {8798, 13157}, {8800, 8905}, {8909, 8966}, {8918, 10218}, {8919, 10217}, {8929, 11581}, {8930, 11582}, {8961, 8963}, {8985, 8990}, {9159, 11639}, {9172, 10162}, {9512, 11061}, {9535, 9566}, {9538, 9642}, {9539, 9641}, {9542, 9691}, {9543, 9690}, {9544, 9704}, {9545, 9703}, {9782, 9809}, {9786, 9815}, {9862, 10583}, {9874, 12872}, {9919, 13203}, {9964, 12528}, {10037, 10832}, {10038, 10874}, {10040, 10925}, {10041, 10926}, {10046, 10831}, {10047, 10873}, {10048, 10923}, {10049, 10924}, {10053, 12185}, {10054, 12351}, {10058, 12764}, {10059, 12860}, {10060, 12950}, {10061, 12951}, {10062, 12952}, {10063, 12836}, {10064, 12954}, {10065, 12374}, {10066, 12956}, {10067, 12958}, {10068, 12959}, {10069, 12184}, {10070, 12350}, {10074, 12763}, {10075, 12859}, {10076, 12940}, {10077, 12941}, {10078, 12942}, {10079, 12837}, {10080, 12944}, {10081, 12373}, {10082, 12946}, {10083, 12948}, {10084, 12949}, {10085, 12678}, {10086, 13183}, {10087, 13274}, {10088, 12904}, {10089, 13182}, {10090, 13273}, {10091, 12903}, {10168, 11645}, {10187, 12816}, {10188, 12817}, {10266, 12919}, {10278, 10279}, {10311, 10316}, {10312, 10317}, {10524, 10530}, {10528, 10596}, {10529, 10597}, {10546, 11464}, {10575, 11381}, {10584, 10785}, {10585, 10786}, {10586, 10805}, {10587, 10806}, {10628, 11557}, {10634, 10641}, {10635, 10642}, {10733, 12121}, {10797, 10802}, {10798, 10801}, {10984, 11550}, {11236, 12513}, {11264, 13366}, {11392, 11399}, {11393, 11398}, {11411, 11431}, {11425, 12118}, {11429, 12428}, {11449, 12278}, {11455, 12279}, {11456, 11457}, {11501, 11508}, {11502, 11507}, {11536, 12234}, {11576, 12363}, {11649, 12061}, {11671, 13512}, {11746, 12236}, {11754, 11755}, {11763, 11764}, {11772, 11773}, {11781, 11782}, {11869, 11879}, {11870, 11880}, {11871, 11877}, {11872, 11878}, {11905, 11913}, {11906, 11912}, {11930, 11953}, {11931, 11954}, {11932, 11951}, {11933, 11952}, {12099, 12827}, {12308, 12317}, {12309, 12318}, {12310, 12319}, {12311, 12320}, {12312, 12321}, {12316, 12325}, {12383, 12902}, {12384, 13115}, {12494, 13234}, {12599, 12864}, {12600, 13089}, {12613, 12621}, {12614, 12622}, {12615, 12623}, {12624, 13249}, {12849, 13126}, {12945, 13117}, {12947, 13129}, {12955, 13116}, {12957, 13128}, {13023, 13025}, {13024, 13026}, {13039, 13051}, {13040, 13052}, {13219, 13310}, {13296, 13312}, {13297, 13311}, {13348, 13570}, {13507, 13508}

▸ Functional Relations72 relations

• X₅ = midpoint of X_i and X_j for these (i,j): (1, 355), (2, 381), (3, 4), (11, 119), (20, 382), (68, 155), (110, 265), (113, 125), (114, 115), (116, 118), (117, 124), (122, 133), (127, 132), (128, 137), (129, 130), (131, 136), (399, 3448)

• X₅ = reflection of X_i in X_j for these (i,j): (2, 547), (3, 140), (4, 546), (20, 548), (52, 143), (549, 2), (550, 3), (1263, 137), (1353, 6), (1385, 1125), (1483, 1), (1484, 11)

• X₅ = isogonal conjugate of X₅₄

• X₅ = isotomic conjugate of X₉₅

• X₅ = circumcircle-inverse of X₂₀₇₀

• X₅ = orthocentroidal-circle-inverse of X₃

• X₅ = de-Longchamps-circle-inverse of anticomplement of X₃₇₉₄₃

• X₅ = circle-O(PU(4))-inverse of X₃₇₉₆₉

• X₅ = complement of X₃

• X₅ = anticomplement of X₁₄₀

• X₅ = complementary conjugate of X₃

• X₅ = eigencenter of anticevian triangle of X₅₂₃

• X₅ = X_i-Ceva conjugate of X_j for these (i,j): (2, 216), (4, 52), (110, 523), (264, 324), (265, 30), (311, 343), (324, 53)

• X₅ = cevapoint of X_i and X_j for these (i,j): (3, 195), (51, 216)

• X₅ = X_i-cross conjugate of X_j for these (i,j): (51, 53), (54, 2121), (216, 343), (233, 2)

• X₅ = crosspoint of X_i and X_j for these (i,j): (2, 264), (311, 324)

• X₅ = crosssum of X_i and X_j for these (i,j): (3, 1147), (6, 184)

• X₅ = crossdifference of every pair of points on line X₅₀X₆₄₇

• X₅ = X₁-aleph conjugate of X₁₀₄₈

• X₅ = radical center of Stammler circles

• X₅ = center of inverse-in-circumcircle-of-tangential-circle

• X₅ = harmonic center of 1st & 2nd Hutson circles

• X₅ = homothetic center of circumorthic triangle and 2nd isogonal triangle of X₄; see X₃₆

• X₅ = X₃-of-X₄-Brocard-triangle

• X₅ = X₄-of-Schroeter-triangle

• X₅ = X₅-of-Fuhrmann-triangle

• X₅ = X₅-of-complement-of-excentral-triangle (or extraversion triangle of X₁₀)

• X₅ = X₁₁₄-of-1st-Brocard-triangle

• X₅ = X₁₄₃-of-excentral-triangle

• X₅ = X₁₅₆-of-intouch-triangle

• X₅ = X₁₅₁₁-of-orthocentroidal-triangle

• X₅ = bicentric sum of PU(i) for these i: 5, 7, 38, 65, 173

• X₅ = midpoint of PU(i) for these i: 5, 7, 38

• X₅ = trilinear product of PU(69)

• X₅ = PU(65)-harmonic conjugate of X₃₅₁

• X₅ = perspector of circumconic centered at X₂₁₆

• X₅ = center of circumconic that is locus of trilinear poles of lines passing through X₂₁₆

• X₅ = trilinear pole of line X₂₀₈₁X₂₆₀₀

• X₅ = pole wrt polar circle of trilinear polar of X₂₇₅ (line X₁₈₆X₅₂₃)

• X₅ = X₄₈-isoconjugate (polar conjugate) of X₂₇₅

• X₅ = X₂₅₂-isoconjugate of X₂₉₆₄

• X₅ = homothetic center of medial triangle and Euler triangle

• X₅ = homothetic center of ABC and the triangle obtained by reflecting X₃ in the points A, B, C

• X₅ = radical center of the Stammler circles

• X₅ = centroid of {A, B, C, X₄}

• X₅ = antigonal image of X₁₂₆₃

• X₅ = crosspoint of X₆₂₇ and X₆₂₈ wrt both the excentral and anticomplementary triangles

• X₅ = intersection of tangents to Evans conic at X₁₅ and X₁₆

• X₅ = polar-circle-inverse of X₁₈₆

• X₅ = inverse-in-{circumcircle, nine-point circle}-inverter of X₂₃

• X₅ = inverse-in-Kiepert-hyperbola of X₃₉

• X₅ = inverse-in-Steiner-inellipse of X₂₉₇

• X₅ = {X₂₀₀₉,X₂₀₁₀}-harmonic conjugate of X₃₉

• X₅ = {X₂₄₅₄,X₂₄₅₅}-harmonic conjugate of X₂₉₇

• X₅ = perspector of medial triangles of ABC, orthic and half-altitude triangles

• X₅ = X₆-isoconjugate of X₂₁₆₇

• X₅ = orthic-isogonal conjugate of X₅₂

• X₅ = Thomson-isogonal conjugate of X₆₀₃₀

• X₅ = X₁-of-submedial triangle if ABC is acute

• X₅ = harmonic center of circumcircles of Euler and anti-Euler triangles

• X₅ = perspector of Feuerbach triangle and cross-triangle of ABC and Feuerbach triangle

• X₅ = Kosnita(X₄,X₂) point

• X₅ = Kosnita(X₄,X₃) point

• X₅ = Kosnita(X₄,X₂₀) point

• X₅ = X₄-of-Ehrmann-mid-triangle

• X₅ = homothetic center of Ehrmann vertex-triangle and Kosnita triangle

• X₅ = homothetic center of Ehrmann side-triangle and circumorthic triangle

• X₅ = perspector of Ehrmann mid-triangle and submedial triangle

• X₅ = Ehrmann-side-to-orthic similarity image of X₄

• X₅ = Johnson-to-Ehrmann-mid similarity image of X₃

• X₅ = excentral-to-ABC functional image of X₃

• X₅ = QA-P32 center (Centroid of the Circumcenter Quadrangle) of quadrangle ABCX₄ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/60-qa-p32.html)

Notes

As a point on the Euler line, X₅ has Shinagawa coefficients \((1, 1)\).

Dan Pedoe, Circles: A Mathematical View, Mathematical Association of America, 1995.

Let P_a be the parabola with focus A and directrix BC. Let L_a be the polar of X₃ wrt P_a. Define L_b and L_c cyclically. Let A' = L_b∩L_c, and define B' and C; cyclically. Then X₅ = X₅₉₇-of-A'B'C'. (Randy Hutson, December 10, 2016)

Let A'B'C' be the half-altitude triangle. Let A″ be the trilinear pole, wrt A'B'C', of line BC, and define B″ and C″ cyclically. The lines A’″, B'B″, C'C″ concur in X₅. (Randy Hutson, December 10, 2016)

Let A' be the nine-point center of BCX₁₃, and define B' and C' cyclically. Let A″ be the nine-point center of BCX₁₄, and define B″ and C″ cyclically. The lines A’″, B'B″, C'C″ concur in X₅. (Randy Hutson, December 10, 2016)

Let Eu_aEu_bEu_c be the Euler triangle. Let A″ be the centroid of AEu_bEu_c, and define B″ and C″ cyclically. The lines A’″, Eu_bB″, Eu_cC″ concur in X₅. (Randy Hutson, December 10, 2016)

Let A'B'C' be any equilateral triangle inscribed in the circumcircle of ABC. The Simson lines of A', B', C' form an equilateral triangle with center X₅. If A'B'C' is the circumtangential triangle, the Simson lines of A', B', C' concur in X₅. (Randy Hutson, December 10, 2016) Let O_A be the circle centered at the A-vertex of the Steiner triangle and passing through A; define O_B and O_C cyclically. X₅ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let O_A be the circle centered at the A-vertex of the 1st excosine triangle and passing through A; define O_B and O_C cyclically. X₅ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let Eu_aEu_bEu_c be the Feuerbach triangle. Let A″ be the cevapoint of Eu_b and Eu_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₅. (Randy Hutson, July 20, 2016)

Let Ref_aRef_bRef_c be the reflection triangle. Let A″ be the trilinear pole of line Ref_bRef_c, and define B″and C″ cyclically. The lines AA″, BB″, CC″ concur in X₅. (Randy Hutson, July 20, 2016)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X₅ = X₆₁₄₆-of-A'B'C'.

Let A'B'C' be the cevian triangle of X₅. Let A″ be X₅-of-AB'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₅. (Randy Hutson, June 27, 2018)

Let N_a, N_b, N_c be the nine-point centers of BCF, CAF, ABF, resp., where F = X₁₃. Let N_a', N_b', N_c' be the nine-point centers of BCF', CAF', ABF', resp., where F' = X₁₄. The lines N_aN_a', N_bN_b', N_cN_c' concur in X₅. (Randy Hutson, June 27, 2018)

Let N_a, N_b, N_c be the nine-point centers of BCX, CAX, ABX, resp., where X = X₁₇. Let N_a', N_b', N_c' be the nine-point centers of BCX', CAX', ABX', resp., where X' = X₁₈. The lines N_aN_a', N_bN_b', N_cN_c' concur in X₅. (Randy Hutson, June 27, 2018)

X₅ is the {X₂,X₄}-harmonic conjugate of X₃.