X₁₀

SPIEKER CENTER

The Spieker circle is the incircle of the medial triangle; its center, X₁₀, is the centroid of the perimeter of ABC.

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

X₁₀ (SPIEKER CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Circle: Spieker Circle (spieker_circle) — Incircle of the medial triangle △M_a M_b M_c; centered at X₁₀ SPIEKER CENTER

• Concurrence Lines: M_a–X₁₀, M_b–X₁₀, M_c–X₁₀ concurring at X₁₀

Trilinears: \(bc(b + c) : ca(c + a) : ab(a + b)\)

Barycentrics: \(b + c : c + a : a + b\)

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

Search Value (6, 9, 13): \(3.098898934005\)

▸ Alternate Coordinates7 forms

• Trilinears: \(1/(r \cos A - s \sin A) : :\)

• Trilinears: \(\csc(A - U) : :\), U as at X₅₇₂ and X₅₇₃

• Trilinears: \((\cos B + \cos C)/(1 - \cos A) : :\)

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

• Trilinears: \(|AP(1)| + |AU(1)| : :\)

• Trilinears: \((r/R) - 2 \sin B \sin C : :\)

• Barycentrics: semi-major axis of A-Soddy ellipse : :

▸ Linear Combinations1 combination

• X₁₀ = X₁ - \(3\) X₂ = \(3\) X₂ + X₈ = X₇ + X₈ + \(2\) X₉

X₁₀ lies on the Kiepert hyperbola.

Related Collection 1 Topic Guides

T_3: Complement & Anticomplement • T_7: The Nagel Line • T_12: The Fuhrmann & Spieker Central Lines • T_17: The Spieker Circle • T_20: Medial & Anticomplementary Triangles

▸ Incident Central Lines85 lines

{1, 2}, {3, 197}, {4, 9}, {5, 517}, {6, 1377}, {11, 121}, {12, 65}, {20, 165}, {21, 35}, {28, 1891}, {29, 1794}, {31, 964}, {33, 406}, {34, 475}, {36, 404}, {37, 594}, {38, 596}, {39, 730}, {44, 752}, {46, 63}, {55, 405}, {56, 474}, {57, 388}, {58, 171}, {69, 969}, {75, 76}, {81, 1224}, {82, 83}, {86, 319}, {87, 979}, {92, 1838}, {98, 101}, {106, 1222}, {116, 120}, {117, 123}, {119, 124}, {140, 214}, {141, 142}, {150, 1282}, {153, 1768}, {158, 318}, {182, 1678}, {190, 671}, {191, 267}, {201, 225}, {219, 965}, {227, 1214}, {235, 1902}, {255, 1771}, {257, 1581}, {261, 1326}, {274, 291}, {307, 1254}, {321, 756}, {348, 1323}, {391, 1743}, {407, 1867}, {427, 1829}, {429, 1824}, {480, 954}, {485, 1686}, {486, 1685}, {497, 1697}, {514, 764}, {535, 1155}, {537, 1086}, {626, 760}, {631, 944}, {632, 1483}, {750, 1150}, {774, 1736}, {775, 801}, {846, 1247}, {894, 1046}, {908, 994}, {962, 1695}, {1018, 1334}, {1074, 1735}, {1146, 1212}, {1482, 1656}, {1587, 1703}, {1588, 1702}, {1762, 1782}, {1828, 1883}, {1900, 1904}

▸ Functional Relations82 relations

• X₁₀ = midpoint of X_i and X_j for these (i,j): (1, 8), (3, 355), (4, 40), (6, 3416), (10, 3421), (55, 3419), (65, 72), (80, 100), (2948, 3448)

• X₁₀ = reflection of X_i in X_j for these (i,j): (1, 1125), (551, 2), (946, 5), (1385, 140)

• X₁₀ = isogonal conjugate of X₅₈

• X₁₀ = isotomic conjugate of X₈₆

• X₁₀ = circumcircle-inverse of X₁₃₂₄

• X₁₀ = nine-point-circle-inverse of X₃₈₁₄

• X₁₀ = Conway-circle-inverse of X₃₈₄₇₆

• X₁₀ = complement of X₁

• X₁₀ = anticomplement of X₁₁₂₅

• X₁₀ = complementary conjugate of X₁₀

• X₁₀ = anticomplementary conjugate of X₂₈₉₁

• X₁₀ = X₁₅₃₁₉-complementary conjugate of X₃₂₇₆₇

• X₁₀ = radical center of the excircles.

• X₁₀ = radical center of extraversions of Conway circle

• X₁₀ = radical center of the polar circles of triangles BCI, CAI, ABI

• X₁₀ = X₂₀-of-3rd-Euler-triangle

• X₁₀ = X₄-of-4th-Euler-triangle

• X₁₀ = perspector of ABC and the tangential triangle of the Feuerbach triangle

• X₁₀ = X₂-Hirst inverse of X₆₅₄₂

• X₁₀ = inverse-in-Steiner-circumellipse of X₆₅₄₂

• X₁₀ = SS(a→a’) of X₅, where J_aJ_bJ_c is the excentral triangle (barycentric substitution)

• X₁₀ = orthocenter of X₂X₄X₄₀₄₉

• X₁₀ = midpoint of PU(10)

• X₁₀ = bicentric sum of PU(i) for these i: 10, 66

• X₁₀ = PU(66)-harmonic conjugate of X₃₅₁

• X₁₀ = crosssum of X_i and X_j for these (i,j): (6, 31), (56, 603)

• X₁₀ = crossdifference of every pair of points on line X₆₄₉X₈₃₄

• X₁₀ = X_i-beth conjugate of X_j for these (i,j): (8, 10), (10, 65), (100, 73), (318, 225), (643, 35), (668, 349)

• X₁₀ = radical trace of Bevan circle and anticomplementary circle

• X₁₀ = insimilicenter of Bevan circle and anticomplementary circle

• X₁₀ = insimilicenter of nine-point circle and Apollonius circle

• X₁₀ = X_i-Ceva conjugate of X_j for these (i,j): (2, 37), (8, 72), (75, 321), (80, 519), (100, 522), (313, 306)

• X₁₀ = cevapoint of X_i and X_j for these (i,j): (1, 191), (6, 199), (12, 201), (37, 210), (42, 71), (65, 227)

• X₁₀ = X_i-cross conjugate of X_j for these (i,j): (37, 226), (71, 306), (191, 502), (201, 72)

• X₁₀ = crosspoint of X_i and X_j for these (i,j): (2, 75), (8, 318)

• X₁₀ = centroid of ABCX₈

• X₁₀ = Kosnita(X₈,X₂) point

• X₁₀ = X₅₇₈-of-2nd-extouch-triangle

• X₁₀ = X₃₈₉-of-excentral triangle

• X₁₀ = X₁₂₅-of-Fuhrmann triangle

• X₁₀ = perspector of ABC and triangle formed from orthocenters of J_aBC, J_bCA, J_cAB, where J_a, J_b, J_c are excenters

• X₁₀ = perspector of circumconic centered at X₃₇

• X₁₀ = center of circumconic that is locus of trilinear poles of lines passing through X₃₇

• X₁₀ = trilinear pole of line X₅₂₃X₆₆₁ (the polar of X₂₇ wrt polar circle)

• X₁₀ = pole wrt polar circle of trilinear polar of X₂₇ (line X₂₄₂X₅₁₄)

• X₁₀ = X₄₈-isoconjugate (polar conjugate)-of-X₂₇

• X₁₀ = X₆-isoconjugate of X₈₁

• X₁₀ = X₇₅-isoconjugate of X₂₂₀₆

• X₁₀ = X₁₁₀₁-isoconjugate of X₃₁₂₀

• X₁₀ = X₁-of-X₁-Brocard triangle

• X₁₀ = perspector of medial triangle and Ayme triangle

• X₁₀ = homothetic center of Ayme triangle and anticevian triangle of X₃₇

• X₁₀ = perspector of Ayme triangle and Danneels-Bevan triangle

• X₁₀ = X₁-of-Danneels-Bevan-triangle

• X₁₀ = homothetic center of medial triangle and Danneels-Bevan triangle

• X₁₀ = homothetic center of ABC and anticomplementary triangle of Danneels-Bevan triangle

• X₁₀ = {X₂,X₈}-harmonic conjugate of X₁

• X₁₀ = inverse-in-polar-circle of X₂₄₂

• X₁₀ = inverse-in-{circumcircle, nine-point circle}-inverter of X₅₂₀₅

• X₁₀ = inverse-in-Steiner-inellipse of X₃₉₁₂

• X₁₀ = inverse-in-Feuerbach-hyperbola of X₃₀₅₇

• X₁₀ = perspector of Feuerbach and Apollonius triangles

• X₁₀ = perspector of symmedial triangles of Feuerbach and Apollonius triangles

• X₁₀ = perspector of circumsymmedial triangles of Feuerbach and Apollonius triangles

• X₁₀ = perspector of tangential triangles of Feuerbach and Apollonius triangles

• X₁₀ = X₂₁₄-of-inner-Garcia-triangle

• X₁₀ = Cundy-Parry Phi transform of X₁₃₄₇₈

• X₁₀ = Cundy-Parry Psi transform of X₅₇₃

• X₁₀ = perspector of Ayme and 4th Euler triangles

• X₁₀ = barycentric product X₁₀₁*X₈₅₀

• X₁₀ = perspector of Gemini triangle 12 and cross-triangle of ABC and Gemini triangle 12

• X₁₀ = perspector of ABC and cross-triangle of ABC and Gemini triangle 15

• X₁₀ = trilinear product of vertices of Gemini triangle 15

• X₁₀ = homothetic center of Ayme triangle and Gemini triangle 16

• X₁₀ = center of the {ABC, Gemini 18}-circumconic

• X₁₀ = Gemini-triangle-19-to-ABC parallelogic center

• X₁₀ = centroid of Gemini triangle 20

• X₁₀ = perspector of ABC and cross-triangle of ABC and Gemini triangle 25

• X₁₀ = perspector of ABC and Gemini triangle 26

• X₁₀ = perspector of Gemini triangle 39 and cross-triangle of ABC and Gemini triangle 39

• X₁₀ = excentral-to-ABC barycentric image of X₃

• X₁₀ = incentral-to-ABC barycentric image of X₁

Notes

Let A' be the intersection of these three lines: the perpendicular from midpoint of CA to line BX₁, the perpendicular from midpoint of AB to line CX₁, the perpendicular from midpoint of AX₁ to line BC, and define B' and C' cyclically. The orthocenter of A'B'C' is X₁₀, and X₁₀ is also the perspector of A'B'C' and the medial triangle. Note that A'B'C' is the complement of the excentral triangle, and the extraversion triangle of X₁₀. (Randy Hutson, December 2, 2017)

A construction of X₁₀ is given at 24163. (Antreas Hatzipolakis, August 29, 2016)

Let J_aJ_bJ_c be the excentral triangle. X₁₀ is the radical center of the polar circles of triangles J_aBC, J_bCA, J_cAB. (Randy Hutson, July 31 2018)

Let A₂₀B₂₀C₂₀ be the Gemini triangle 20. Let L_A be the line through A₂₀ parallel to BC, and define L_B and L_C cyclically. Let A'₂₀ = L_B∩L_C, and define B'₂₀ and C'₂₀ cyclically. Triangle A'₂₀B'₂₀C'₂₀ is homothetic to ABC at X₁₀. (Randy Hutson, November 30, 2018)

Let O_A be the circle centered at the A-excenter and passing through A; define O_B and O_C cyclically. X₁₀ 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 hexyl triangle and passing through A; define O_B and O_C cyclically. X₁₀ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let

• B_a = reflection of A with in the external angular bisector of B, and define C_b and A_c cyclically;

• C_a = reflection of A with in the external angular bisector of C, and define A_b and B_c cyclically;

• V_a = CB_c∩BC_b, and define V_b and V_c cyclically.

The triangle V_aV_bV_c is perspective to ABC, and the perspector is X₁₀. (Dasari Naga Vijay Krishna, April 19, 2021)

X₁₀ is the {X₁,X₂}-harmonic conjugate of X₁₁₂₅. X₁₀ is the internal center of similitude of the Apollonius and nine-points circles.

Let A'B'C' be the 2nd extouch triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁₀. Also, let A″B″C″ be the 1st circumperp triangle. The Simson lines of A″, B″, C″ concur in X₁₀. (Randy Hutson, November 18, 2015)