X2

CENTROID

X2 is the point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X2 to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X2 and also balances atop any knife-edge that passes through X2. The triangles BXC, CXA, AXB have equal areas if and only if X = X2.
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A M_a ]: A-Median [ B M_b ]: B-Median [ C M_c ]: C-Median { 3, 4 }: Euler Line { 1, 8 }: Nagel Line { 7, 9 }: Centroid–Gergonne Line { 13, 16 }: Central Line L(13, 16) { 14, 15 }: Central Line L(14, 15) 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_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1 = INCENTER (0.5714, 0.2333) X_8 = NAGEL POINT (0.4694, 0.1333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_9 = MITTENPUNKT (0.5110, 0.1615) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_16 = 2nd ISODYNAMIC POINT (1.8453, 2.3239) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_2 = CENTROID (0.5374, 0.2000) A B C Ma Mb Mc X3 X4 X1 X8 X7 X9 X13 X16 X14 X15 X5 X10 X2

X2 (CENTROID) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Derived Triangle: Medial Triangle △MaMbMc — vertices Ma, Mb, Mc (Formed by the midpoints of sides BC, CA, AB)

  • Concurrence Lines: A–Ma, B–Mb, C–Mc concurring at X2

  • Central Lines: {3, 4} (Euler line)

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

Trilinears: \(1/a : 1/b : 1/c\)
Barycentrics: \(1 : 1 : 1\)
Tripolars: \(\sqrt{2(b^{2} + c^{2}) - a^{2}} : :\)
Search Value (6, 9, 13): \(2.629368792489\)
▸ Alternate Coordinates8 forms

  • Trilinears: \(bc : ca : ab\)

  • Trilinears: \(\csc A : \csc B : \csc C\)

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

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

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

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

  • Trilinears: \(\tan(A/2) + \cot(A/2) : :\)

  • Trilinears: \(1 + \csc A/2 \sin B/2 \sin C/2 : :\)

▸ Linear Combinations1 combination

X2 lies on the Parry circle, Lucas cubic, Thomson cubic.

Related Collection 1 Topic Guides

T_1: Isogonal Conjugate • T_2: Isotomic Conjugate • T_3: Complement & Anticomplement • T_4: Harmonic Conjugate & Cross-Ratio • T_6: The Euler Line • T_7: The Nagel Line • 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_17: The Spieker Circle • T_19: The Brocard & Orthocentroidal Circles • T_20: Medial & Anticomplementary Triangles • T_22: Intouch (Contact) & Extouch Triangles • T_24: Equilateral, Isogonic & Napoleon Triangles

▸ Incident Central Lines162 lines

{1, 8}, {3, 4}, {6, 69}, {7, 9}, {11, 55}, {12, 56}, {13, 16}, {14, 15}, {17, 62}, {18, 61}, {19, 534}, {31, 171}, {32, 83}, {33, 1040}, {34, 1038}, {35, 1479}, {36, 535}, {37, 75}, {38, 244}, {39, 76}, {40, 946}, {44, 89}, {45, 88}, {51, 262}, {52, 1216}, {54, 68}, {58, 540}, {65, 959}, {66, 206}, {71, 1246}, {72, 942}, {74, 113}, {77, 189}, {80, 214}, {85, 241}, {92, 273}, {94, 300}, {95, 97}, {98, 110}, {99, 111}, {101, 116}, {102, 117}, {103, 118}, {104, 119}, {106, 121}, {107, 122}, {108, 123}, {109, 124}, {112, 127}, {128, 1141}, {129, 1298}, {130, 1303}, {131, 1300}, {133, 1294}, {136, 925}, {137, 930}, {154, 1503}, {165, 516}, {169, 1763}, {174, 236}, {176, 1659}, {178, 188}, {187, 316}, {196, 653}, {201, 1393}, {210, 354}, {216, 232}, {220, 1170}, {222, 651}, {231, 1273}, {242, 1851}, {243, 1857}, {252, 1166}, {253, 1073}, {254, 847}, {257, 1432}, {261, 593}, {265, 1511}, {271, 1034}, {272, 284}, {280, 318}, {283, 580}, {290, 327}, {292, 334}, {294, 949}, {308, 702}, {311, 570}, {314, 941}, {319, 1100}, {322, 1108}, {330, 1107}, {341, 1219}, {351, 804}, {355, 944}, {360, 1115}, {366, 367}, {371, 486}, {372, 485}, {392, 517}, {476, 842}, {480, 1223}, {489, 1132}, {490, 1131}, {495, 956}, {496, 1058}, {514, 1022}, {523, 1649}, {525, 1640}, {561, 716}, {568, 1154}, {572, 1746}, {573, 1730}, {578, 1092}, {585, 1336}, {586, 1123}, {588, 1504}, {589, 1505}, {594, 1255}, {647, 850}, {648, 1494}, {650, 693}, {664, 1121}, {668, 1015}, {670, 1084}, {689, 733}, {743, 789}, {799, 873}, {812, 1635}, {846, 1054}, {914, 1442}, {918, 1638}, {927, 1566}, {954, 1260}, {1073, 1249}, {968, 1738}, {1000, 1145}, {1043, 1834}, {1060, 1870}, {1074, 1785}, {1076, 1838}, {1089, 1224}, {1093, 1217}, {1124, 1378}, {1143, 1489}, {1155, 1836}, {1171, 1509}, {1186, 1207}, {1257, 1265}, {1284, 1403}, {1335, 1377}, {1340, 1349}, {1341, 1348}, {1500, 1574}, {1501, 1691}, {1672, 1681}, {1673, 1680}, {1674, 1679}, {1675, 1678}, {1697, 1706}, {3343, 3344}, {3349, 3350}, {3351, 3352}

▸ Functional Relations142 relations

Notes

As a point on the Euler line, X2 has Shinagawa coefficients \((1, 0)\).

X2 is the centroid of the set of 3 vertices, the centroid of the triangle including its interior, but not the centroid of the triangle without its interior; that centroid is X10.

X2 is the identity of the group of triangle centers under “barycentric multiplication” defined by

\((x : y : z) \cdot (u : v : w) = xu : yv : zw\).

X2 is the unique point X (as a function of a,b,c) for which the vector-sum XA + XB + XC is the zero vector.

The Parry isodynamic circle is here introduced as the circle centered at X2502 that passes through the isodynamic points, X15 and X16. This circle is orthogonal to both the circumcircle and Parry circle. (Randy Hutson, February 10, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. A'B'C' is homothetic to the midheight triangle at X2. (Randy Hutson, January 29, 2018)

Let JaJbJc be the excentral triangle. Let Oa be the Ja-McCay circle of triangle JaBC, and define Ob, Oc cyclically. X2 is the radical center of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X2 is the unique point that is the symmedian point of its antipedal triangle. (Randy Hutson, August 19, 2019)

Let A'B'C' be the midheight triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X2. (Randy Hutson, October 8, 2019)

In the plane of a triangle ABC, let

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

  • A' = the reflection of A with in Oa, and define B' and C' cyclically.

The triangle A'B'C' is perspective to ABC, and the perspector is X2. (Dasari Naga Vijay Krishna, April 19, 2021)

Let Na = reflection of ninepoint center(N) wrt to BC, define Nb, Nc cyclically; then NaNbNc is perspective to ABC, and the perspector is X2. (Dasari Naga Vijay Krishna, June 8, 2021)

Let O be a point(not necessarily X3), and let 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 X2-of-PAB is a circle C'=(O',R') whose center O' lies on OP. The radius R', independent of P, is given by R'=\(\sqrt{2+\sqrt{2}}\)(R/3). Figure. (Dan Reznik, December 11, 2021)

In the plane of a triangle ABC, let P be a point, and let

  • A'B'C' = pedal triangle of P;

  • O' = circumcenter of A'B'C';

  • A″ = reflection of A' in O', and define B″ and C″ cyclically.

The triangles ABC and A″B″C″ are perspective, and their pespector is named here the pedal antipodal perspector of P. (Randy Hutson, Hyacinthos 20403, Nov. 21, 2011).

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