X3

CIRCUMCENTER

X3 is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X3. This common distance is the radius of the circumcircle, which passes through vertices A, B, C. Called the circumradius, it is given by \(R = a/(2 \sin A) = abc/(4 \operatorname{area}(ABC))\).
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC [ X_3 M_a ]: Perpendicular Bisector of BC [ X_3 M_b ]: Perpendicular Bisector of CA [ X_3 M_c ]: Perpendicular Bisector of AB { 2, 4 }: Euler Line { 6, 15 }: Brocard Axis { 13, 17 }: Central Line L(13, 17) { 14, 18 }: Central Line L(14, 18) { 1, 35 }: OI Line (Incenter–Circumcenter Line) { 1, 40 }: OI Line (Incenter–Circumcenter Line) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5374, 0.2000) X_4 = ORTHOCENTER (0.6122, 0.3958) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_17 = 1st NAPOLEON POINT (0.5666, 0.2383) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_18 = 2nd NAPOLEON POINT (0.4434, 0.1317) X_1 = INCENTER (0.5714, 0.2333) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_40 = BEVAN POINT (0.4286, -0.0292) X_3 = CIRCUMCENTER (0.5000, 0.1021) A B C Ma Mb Mc X2 X4 X6 X15 X13 X17 X14 X18 X1 X35 X5 X40 X3

X3 (CIRCUMCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Auxiliary Points: Ma, Mb, Mc — Side midpoints of BC, CA, AB

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

  • Concurrence Lines (Perpendicular Bisectors): X3–Ma, X3–Mb, X3–Mc concurring at X3

  • Central Lines: {2, 4} (Euler line)

  • Related Centers: X2 (CENTROID), X4 (ORTHOCENTER)

Trilinears: \(\cos A : \cos B : \cos C\)
Barycentrics: \(\sin 2A : \sin 2B : \sin 2C\)
Tripolars: \(1 : 1 : 1\)
Search Value (6, 9, 13): \(6.782362894196\)
▸ Alternate Coordinates4 forms

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

  • Barycentrics: \(\tan B + \tan C : \tan C + \tan A: \tan A + \tan B\)

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

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

X3 lies on the Thomson cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, the McCay cubic, then Darboux quintic.

Related Collection 1 Topic Guides

T_1: Isogonal Conjugate • T_3: Complement & Anticomplement • T_4: Harmonic Conjugate & Cross-Ratio • T_5: Circle Inversion • T_6: The Euler Line • T_8: The Brocard Axis • T_10: The OI Line (Incenter–Circumcenter Axis) • T_11: The Soddy Line • T_12: The Fuhrmann & Spieker Central Lines • T_13: The Isogonic & Napoleon Central Lines • T_15: The Circumcircle • T_16: The Nine-Point (Feuerbach) Circle • T_18: The Apollonian Circles • T_19: The Brocard & Orthocentroidal Circles • T_20: Medial & Anticomplementary Triangles • T_21: Orthic & Tangential Triangles • T_23: Incentral & Excentral Triangles • T_24: Equilateral, Isogonic & Napoleon Triangles

▸ Incident Central Lines500 lines

{1, 35}, {2, 4}, {6, 15}, {7, 943}, {8, 100}, {9, 84}, {10, 197}, {11, 499}, {12, 498}, {13, 17}, {14, 18}, {19, 1871}, {31, 601}, {33, 1753}, {34, 1465}, {37, 975}, {38, 976}, {41, 218}, {42, 967}, {43, 5247}, {47, 1399}, {48, 71}, {49, 155}, {51, 3527}, {54, 97}, {60, 1175}, {63, 72}, {64, 154}, {66, 141}, {67, 542}, {68, 343}, {69, 332}, {73, 212}, {74, 110}, {76, 98}, {77, 1410}, {80, 5445}, {81, 5453}, {83, 262}, {85, 5088}, {86, 1246}, {90, 1898}, {95, 264}, {96, 5392}, {101, 103}, {102, 109}, {105, 277}, {106, 1293}, {107, 1294}, {108, 1295}, {111, 1296}, {112, 1297}, {113, 122}, {114, 127}, {115, 2079}, {119, 123}, {125, 131}, {128, 1601}, {142, 516}, {143, 1173}, {144, 5843}, {145, 1483}, {147, 2896}, {149, 1484}, {158, 243}, {161, 1209}, {164, 3659}, {169, 910}, {172, 2276}, {191, 1768}, {193, 1353}, {194, 385}, {200, 963}, {201, 1807}, {207, 1767}, {214, 2800}, {217, 3289}, {223, 1035}, {225, 1074}, {226, 4292}, {227, 1455}, {230, 2549}, {232, 1968}, {238, 978}, {248, 3269}, {252, 930}, {256, 987}, {269, 939}, {295, 2196}, {296, 820}, {298, 617}, {299, 616}, {302, 621}, {303, 622}, {305, 1799}, {315, 325}, {323, 3431}, {329, 2096}, {345, 1791}, {347, 1119}, {348, 1565}, {351, 2780}, {352, 353}, {356, 3278}, {358, 6120}, {373, 3066}, {380, 2257}, {388, 495}, {390, 1058}, {392, 3420}, {393, 1217}, {395, 398}, {396, 397}, {476, 477}, {480, 5223}, {485, 590}, {486, 615}, {489, 492}, {490, 491}, {496, 497}, {501, 5127}, {513, 3657}, {518, 3433}, {519, 3654}, {523, 5664}, {524, 5486}, {525, 878}, {528, 3813}, {532, 5859}, {533, 5858}, {539, 3519}, {541, 5642}, {543, 5569}, {551, 3653}, {595, 995}, {604, 2269}, {607, 1951}, {608, 1950}, {609, 5280}, {611, 1469}, {612, 5322}, {613, 1428}, {614, 5310}, {618, 635}, {619, 636}, {623, 629}, {624, 630}, {639, 641}, {640, 642}, {653, 1148}, {659, 2826}, {662, 1098}, {667, 1083}, {669, 1499}, {690, 6334}, {691, 842}, {692, 2807}, {695, 1613}, {732, 6308}, {741, 6010}, {758, 5884}, {759, 6011}, {805, 2698}, {840, 2742}, {843, 2709}, {846, 2944}, {847, 925}, {895, 4558}, {901, 953}, {902, 1201}, {905, 1946}, {915, 2969}, {917, 1305}, {920, 1858}, {927, 2724}, {929, 2723}, {934, 972}, {935, 2697}, {938, 3488}, {945, 1457}, {947, 5399}, {950, 1210}, {951, 1407}, {955, 1170}, {960, 997}, {962, 1621}, {968, 6051}, {974, 5504}, {984, 3497}, {1000, 1476}, {1014, 3945}, {1015, 2241}, {1018, 4513}, {1033, 1249}, {1037, 1066}, {1046, 4650}, {1047, 2636}, {1054, 1283}, {1055, 1334}, {1056, 3600}, {1057, 1450}, {1069, 6238}, {1072, 3011}, {1075, 1941}, {1093, 1105}, {1104, 3752}, {1107, 4386}, {1124, 2066}, {1131, 3316}, {1132, 3317}, {1135, 6121}, {1137, 6122}, {1138, 3471}, {1139, 3370}, {1140, 3397}, {1167, 1413}, {1177, 1576}, {1180, 1627}, {1184, 1194}, {1196, 1611}, {1199, 1994}, {1203, 5313}, {1211, 5810}, {1213, 5816}, {1247, 2640}, {1263, 3459}, {1270, 5874}, {1271, 5875}, {1276, 5240}, {1277, 5239}, {1290, 2687}, {1298, 1303}, {1301, 5897}, {1304, 2693}, {1308, 2717}, {1309, 2734}, {1330, 4417}, {1331, 1797}, {1335, 2067}, {1337, 3489}, {1338, 3490}, {1348, 2040}, {1349, 2039}, {1364, 1795}, {1386, 3941}, {1389, 2320}, {1397, 1682}, {1398, 1870}, {1400, 2268}, {1406, 1464}, {1411, 1772}, {1412, 2213}, {1425, 3561}, {1427, 1448}, {1433, 2188}, {1445, 5728}, {1446, 3188}, {1447, 3673}, {1452, 1905}, {1453, 2999}, {1471, 2293}, {1475, 2280}, {1495, 3426}, {1500, 2242}, {1506, 5475}, {1568, 3521}, {1575, 4426}, {1587, 3068}, {1588, 3069}, {1602, 2550}, {1603, 2551}, {1612, 4000}, {1625, 1987}, {1630, 3197}, {1632, 2790}, {1633, 5698}, {1661, 2883}, {1672, 3238}, {1673, 3237}, {1676, 5403}, {1677, 5404}, {1696, 3731}, {1698, 4413}, {1699, 3624}, {1709, 3683}, {1714, 5721}, {1723, 2264}, {1724, 3216}, {1728, 1864}, {1737, 1837}, {1745, 1935}, {1762, 2939}, {1770, 1836}, {1779, 1780}, {1788, 3486}, {1794, 3173}, {1796, 3690}, {1808, 4173}, {1810, 4587}, {1811, 4571}, {1813, 3270}, {1834, 5292}, {1901, 5747}, {1914, 2275}, {1916, 3406}, {1918, 2274}, {1939, 6181}, {1960, 2821}, {1986, 2904}, {2007, 3235}, {2008, 3236}, {2053, 2108}, {2120, 3463}, {2121, 3482}, {2130, 3343}, {2131, 3350}, {2133, 5670}, {2163, 2334}, {2174, 2911}, {2183, 2267}, {2197, 2286}, {2222, 2716}, {2292, 3724}, {2329, 3501}, {2346, 3296}, {2407, 2452}, {2548, 3815}, {2688, 2690}, {2689, 2695}, {2691, 2752}, {2692, 2758}, {2694, 2766}, {2696, 2770}, {2699, 2703}, {2700, 2702}, {2701, 2708}, {2704, 2711}, {2705, 2712}, {2706, 2713}, {2707, 2714}, {2710, 2715}, {2718, 2743}, {2719, 2744}, {2720, 2745}, {2721, 2746}, {2722, 2747}, {2725, 2736}, {2726, 2737}, {2727, 2738}, {2728, 2739}, {2729, 2740}, {2730, 2751}, {2731, 2757}, {2732, 2762}, {2733, 2765}, {2735, 2768}, {2783, 4436}, {2792, 4655}, {2797, 6130}, {2801, 3678}, {2810, 3939}, {2814, 3960}, {2827, 4491}, {2854, 5505}, {2886, 4999}, {2888, 3448}, {2916, 3456}, {2951, 3646}, {2971, 3563}, {3006, 5300}, {3058, 4309}, {3061, 3496}, {3065, 3467}, {3092, 5413}, {3093, 5412}, {3100, 6198}, {3101, 6197}, {3165, 5669}, {3166, 5668}, {3177, 3732}, {3200, 3205}, {3201, 3206}, {3218, 3418}, {3219, 3876}, {3224, 6234}, {3229, 3360}, {3272, 3334}, {3276, 3280}, {3277, 3282}, {3305, 5927}, {3306, 5439}, {3332, 4648}, {3341, 3347}, {3351, 3354}, {3366, 3391}, {3367, 3392}, {3373, 3387}, {3374, 3388}, {3381, 5402}, {3382, 5401}, {3399, 3407}, {3413, 6178}, {3414, 6177}, {3417, 3869}, {3436, 5552}, {3437, 5224}, {3440, 5674}, {3441, 5675}, {3447, 6328}, {3452, 6259}, {3460, 3465}, {3461, 3483}, {3462, 5667}, {3464, 3466}, {3474, 3485}, {3555, 3870}, {3582, 4330}, {3583, 4324}, {3584, 4325}, {3585, 4316}, {3589, 5480}, {3614, 5326}, {3620, 5921}, {3632, 5288}, {3647, 3652}, {3667, 4057}, {3679, 5258}, {3681, 4420}, {3687, 5814}, {3694, 5227}, {3705, 5015}, {3710, 3977}, {3711, 5531}, {3733, 6003}, {3734, 3934}, {3740, 5302}, {3824, 5715}, {3849, 6232}, {3874, 4973}, {3877, 4881}, {3889, 3957}, {3901, 4880}, {3925, 6253}, {4001, 4101}, {4317, 4995}, {4338, 4870}, {4340, 5323}, {4549, 4846}, {4653, 6176}, {4720, 5372}, {4850, 5262}, {4993, 4994}, {5226, 5714}, {5260, 5818}, {5268, 5345}, {5275, 5277}, {5284, 5550}, {5286, 5305}, {5306, 5319}, {5346, 5355}, {5436, 5437}, {5441, 5442}, {5443, 5444}, {5530, 5725}, {5541, 6264}, {5590, 5594}, {5591, 5595}, {5606, 5951}, {5638, 6141}, {5639, 6142}, {5640, 5643}, {5656, 6225}, {5658, 5811}, {5672, 6191}, {5673, 6192}, {5735, 6173}, {5962, 5963}, {5971, 6031}, {6082, 6093}, {6118, 6250}, {6119, 6251}, {6228, 6230}, {6229, 6231}, {6233, 6323}, {6236, 6325}, {6294, 6295}, {6296, 6298}, {6297, 6299}, {6300, 6302}, {6301, 6303}, {6304, 6306}, {6305, 6307}, {6311, 6313}, {6312, 6314}, {6315, 6317}, {6316, 6318}, {6391, 6461}, {6413, 6458}, {6414, 6457}, {6581, 6582}

▸ Functional Relations138 relations

Notes

As a point on the Euler line, X3 has Shinagawa coefficients \((1, -1)\).

The tangents at vertices of excentral triangle to the McCay cubic K003 concur in X3. Also, the tangents at A, B, C to the orthocubic K006 concur in X3. (Randy Hutson, November 18, 2015)

Let HaHbHc be the cevian triangle of X4. Let A″ be X4-of-AHbHc, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X3. (Randy Hutson, June 27, 2018)

Let P be a point in the plane of ABC. Let P' be the isogonal conjugate of P. Let P″ be the pedal antipodal perspector of P. X3 is the unique point P for which P' = P″. (Randy Hutson, June 27, 2018)

Taking a reference triangle ABC and a variable point P on the plane, P=X3 is the point of maximal area of its pedal triangle when considering all points P inside the circumcircle of ABC. There are points P far away from the circumcircle for which the area of their pedal triangles is much larger. However, if you consider the signed area of the pedal triangle of P (of which sign depends on whether the points are in clockwise or anti-clockwise order), you could just say that the area of the pedal triangle of P is always negative whenever P is outside of the circumcircle so that P=X3 gives the global maximum. (Mark Helman, July 10, 2020)

A slightly similar thing happens regarding the area of the antipedal triangle of P. P=X4 has the smallest area of its antipedal amongst all P in the interior of triangle ABC (when X4 is in this interior). There are points P (on the circumcircle) for which this area goes to 0. However, if we consider the signed area of the antipedal, even though there are still regions of the plane outside of ABC where the signed area is positive, P=X4 gives the smallest area of the antipedal among all P for which this area is positive (this works even when ABC is obtuse, and points close to the circumcircle (on both sides) have negative antipedal area). (Mark Helman, July 10, 2020)

View Extremal Area Pedal and Antipedal Triangles, by Mark Helman, Ronaldo Garcia, and Dan Reznik.

Let T be any one of these trianges: {Aries, X3-Ehrmann, X3-ABC reflections, 3rd pedal of X3, 3rd antipedal of X3, inner-Le Viet An, outer-Le Viet An}. Let OA be the circle centered at the A-vertex of T and passing through A; define OBand OC cyclically. X3 is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let

  • Oa = circle with diameter BC, and define Ob and Oc cyclically;

  • A' = reflection of A in Oa, and define B' and C' cyclically;

  • Ab = BA'∩Oa, and define Bc and Ca cyclically;

  • Ac = CA'∩Oa, and define Ba and Cb cyclically;

  • A1 = BcBa∩CaCb, and define B1 and C1 cyclically.

The triangle A1B1C1 is perspective to ABC, and the perspector is X3. (Dasari Naga Vijay Krishna, April 19, 2021)

Let O be a point (not necessarily X3), and let be a AOB be a fixed-angle sector of a circle C=(O,R), rigidly rotating about center O. Let P be an arbitrary point. The locus of X3-of-PAB is a conic E whose major axis is OP. This conic is an ellipse (resp. hyperbola) if P is interior (resp. exterior) to C. One of its foci is O. Figure (ellipse). Figure (hyperbola). (Dan Reznik, December 10, 2021)

Let A'B'C' be the anticevian triangle of X3, and let Ea be the ellipse passing through A' and having foci B' and C'. Define Eb and Ec cyclically. The 6 major vertices of the three ellipses lie on a circle that is concentric with the circumcircle of A'B'C'. Figure. (Dan Reznik, December 19, 2021)

X3 is the {X2,X4}-harmonic conjugate of X5. If triangle ABC is acute, then X3 is the incenter of the tangential triangle and the Bevan point, X40, of the orthic triangle.