X(10) = SPIEKER CENTER¶
Trilinears
\(bc(b + c) : ca(c + a) : ab(a + b)\)
\(1/(r cos A - s sin A) : :\)
\(csc(A - U) : :, U as at :ref:`X(572) <X(572)>\) and X(573) <X(573)>`
\((cos B + cos C)/(1 - cos A) : :\)
\(1 + 2 csc A/2 sin B/2 sin C/2 : :\)
\(|AP(1)| + |AU(1)| : :\)
\((r/R) - 2 sin B sin C : :\)
Barycentrics
\(b + c : c + a : a + b\)
\(semi-major axis of A-Soddy ellipse : :\)
Notes
Let A’ be the intersection of these three lines: the perpendicular from midpoint of CA to line B:ref:X(1) <X(1)>, the perpendicular from midpoint of AB to line C:ref:X(1) <X(1)>, the perpendicular from midpoint of A:ref:X(1) <X(1)> to line BC, and define B’ and C’ cyclically. The orthocenter of A’B’C’ is X(10), and X(10) 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 X(10). (Randy Hutson, December 2, 2017)
The Spieker circle is the incircle of the medial triangle; its center, X(10), is the centroid of the perimeter of ABC. If you have The Geometer’s Sketchpad, you can view Spieker center. If you have GeoGebra, you can view Spieker center.
A construction of X(10) is given at 24163. (Antreas Hatzipolakis, August 29, 2016)
Let A’B’C’ be the excentral triangle. X(10) is the radical center of the polar circles of triangles A’BC, B’CA, C’AB. (Randy Hutson, July 31 2018)
Let A20B20C20 be the Gemini triangle 20. Let LA be the line through A20 parallel to BC, and define LB and LC cyclically. Let A’20 = LB∩LC, and define B’20 and C’20 cyclically. Triangle A’20B’20C’20 is homothetic to ABC at X(10). (Randy Hutson, November 30, 2018)
Let OA be the circle centered at the A-excenter and passing through A; define OB and OC cyclically. X(10) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)
Let OA be the circle centered at the A-vertex of the hexyl triangle and passing through A; define OB and OC cyclically. X(10) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)
In the plane of a triangle ABC, let Ba = reflection of A with in the external angular bisector of B, and define Cb and Ac cyclically; Ca = reflection of A with in the external angular bisector of C, and define Ab and Bc cyclically; Va = CBc∩BCb, and define Vb and Vc cyclically. The triangle VaVbVc is perspective to ABC, and the perspector is X(10). (Dasari Naga Vijay Krishna, April 19, 2021)
X(10) lies on the Kiepert hyperbola and these 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
X(10) is the {X(1),:ref:X(2) <X(2)>}-harmonic conjugate of X(1125). For a list of other harmonic conjugates of X(10), click Tables at the top of this page. X(10) 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 X(10). Also, let A’’B’’C’’ be the 1st circumperp triangle. The Simson lines of A’’, B’’, C’’ concur in X(10). (Randy Hutson, November 18, 2015)
X(10) = 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)
X(10) = reflection of X(i) in X(j) for these (i,j): (1,1125), (551,2), (946,5), (1385,140)
X(10) = isogonal conjugate of X(58)
X(10) = isotomic conjugate of X(86)
X(10) = circumcircle-inverse of X(1324)
X(10) = nine-point-circle-inverse of X(3814)
X(10) = Conway-circle-inverse of X(38476)
X(10) = complement of X(1)
X(10) = anticomplement of X(1125)
X(10) = complementary conjugate of X(10)
X(10) = anticomplementary conjugate of X(2891)
X(10) = X(15319)-complementary conjugate of X(32767)
X(10) = radical center of the excircles.
X(10) = radical center of extraversions of Conway circle
X(10) = radical center of the polar circles of triangles BCI, CAI, ABI
X(10) = X(20)-of-3rd-Euler-triangle
X(10) = X(4)-of-4th-Euler-triangle
X(10) = perspector of ABC and the tangential triangle of the Feuerbach triangle
X(10) = X(2)-Hirst inverse of X(6542)
X(10) = inverse-in-Steiner-circumellipse of X(6542)
X(10) = SS(a→a’) of X(5), where A’B’C’ is the excentral triangle (barycentric substitution)
X(10) = orthocenter of X(2)X(4)X(4049)
X(10) = midpoint of PU(10)
X(10) = bicentric sum of PU(i) for these i: 10, 66
X(10) = PU(66)-harmonic conjugate of X(351)
X(10) = crosssum of X(i) and X(j) for these (i,j): (6,31), (56,603)
X(10) = crossdifference of every pair of points on line X(649)X(834)
X(10) = X(i)-beth conjugate of X(j) for these (i,j): (8,10), (10,65), (100,73), (318,225), (643,35), (668,349)
X(10) = radical trace of Bevan circle and anticomplementary circle
X(10) = insimilicenter of Bevan circle and anticomplementary circle
X(10) = insimilicenter of nine-point circle and Apollonius circle
X(10) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,37), (8,72), (75,321), (80,519), (100,522), (313,306)
X(10) = cevapoint of X(i) and X(j) for these (i,j): (1,191), (6,199), (12,201), (37,210), (42,71), (65,227)
X(10) = X(i)-cross conjugate of X(j) for these (i,j): (37,226), (71,306), (191,502), (201,72)
X(10) = crosspoint of X(i) and X(j) for these (i,j): (2,75), (8,318)
X(10) = centroid of ABC:ref:X(8) <X(8)>
X(10) = Kosnita(X(8),:ref:X(2) <X(2)>) point
X(10) = X(578)-of-2nd-extouch-triangle
X(10) = X(389)-of-excentral triangle
X(10) = X(125)-of-Fuhrmann triangle
X(10) = perspector of ABC and triangle formed from orthocenters of JaBC, JbCA, JcAB, where Ja, Jb, Jc are excenters
X(10) = perspector of circumconic centered at X(37)
X(10) = center of circumconic that is locus of trilinear poles of lines passing through X(37)
X(10) = trilinear pole of line X(523)X(661) (the polar of X(27) wrt polar circle)
X(10) = pole wrt polar circle of trilinear polar of X(27) (line X(242)X(514))
X(10) = X(48)-isoconjugate (polar conjugate)-of-X(27)
X(10) = X(6)-isoconjugate of X(81)
X(10) = X(75)-isoconjugate of X(2206)
X(10) = X(1101)-isoconjugate of X(3120)
X(10) = X(1)-of-X(1)-Brocard triangle
X(10) = perspector of medial triangle and Ayme triangle
X(10) = homothetic center of Ayme triangle and anticevian triangle of X(37)
X(10) = perspector of Ayme triangle and Danneels-Bevan triangle
X(10) = X(1)-of-Danneels-Bevan-triangle
X(10) = homothetic center of medial triangle and Danneels-Bevan triangle
X(10) = homothetic center of ABC and anticomplementary triangle of Danneels-Bevan triangle
X(10) = {X(2),:ref:X(8) <X(8)>}-harmonic conjugate of X(1)
X(10) = inverse-in-polar-circle of X(242)
X(10) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5205)
X(10) = inverse-in-Steiner-inellipse of X(3912)
X(10) = inverse-in-Feuerbach-hyperbola of X(3057)
X(10) = perspector of Feuerbach and Apollonius triangles
X(10) = perspector of symmedial triangles of Feuerbach and Apollonius triangles
X(10) = perspector of circumsymmedial triangles of Feuerbach and Apollonius triangles
X(10) = perspector of tangential triangles of Feuerbach and Apollonius triangles
X(10) = X(214)-of-inner-Garcia-triangle
X(10) = Cundy-Parry Phi transform of X(13478)
X(10) = Cundy-Parry Psi transform of X(573)
X(10) = perspector of Ayme and 4th Euler triangles
X(10) = barycentric product X(101)
X(10) = perspector of Gemini triangle 12 and cross-triangle of ABC and Gemini triangle 12
X(10) = perspector of ABC and cross-triangle of ABC and Gemini triangle 15
X(10) = trilinear product of vertices of Gemini triangle 15
X(10) = homothetic center of Ayme triangle and Gemini triangle 16
X(10) = center of the {ABC, Gemini 18}-circumconic
X(10) = Gemini-triangle-19-to-ABC parallelogic center
X(10) = centroid of Gemini triangle 20
X(10) = perspector of ABC and cross-triangle of ABC and Gemini triangle 25
X(10) = perspector of ABC and Gemini triangle 26
X(10) = perspector of Gemini triangle 39 and cross-triangle of ABC and Gemini triangle 39
X(10) = excentral-to-ABC barycentric image of X(3)
X(10) = incentral-to-ABC barycentric image of X(1)