X₅₅

INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

X₅₅ is the insimilicenter(circumcircle, incircle).

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

X₅₅ (INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Tangential Triangle △Tan_aTan_bTan_c — vertices Tan_a, Tan_b, Tan_c (Bounded by tangents to the circumcircle at A, B, C)

• Derived Triangle: Intouch (Contact) Triangle △T_aT_bT_c — vertices T_a, T_b, T_c (Contact triangle of Incircle touchpoints on BC, CA, AB)

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

• Circle: Incircle (incircle) — Inscribed circle tangent to sidelines at T_a, T_b, T_c; centered at X₁ INCENTER

• Concurrence Lines: Tan_a–T_a, Tan_b–T_b, Tan_c–T_c concurring at X₅₅

• Central Lines: {1, 3}

• Related Centers: X₃ (CIRCUMCENTER), X₁ (INCENTER)

Trilinears: \(a(b + c - a) : b(c + a - b) : c(a + b - c)\)

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

Search Value (6, 9, 13): \(2.635469416137\)

▸ Alternate Coordinates8 forms

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

• Trilinears: \(\cos^{2}(A/2) : \cos^{2}(B/2) : \cos^{2}(B/2)\)

• Trilinears: \(\tan(B/2) + \tan(C/2) : \tan(C/2) + \tan(A/2) : \tan(A/2) + \tan(B/2)\)

• Trilinears: \(a(a - s) : b(b - s) : c(c - s)\)

• Trilinears: \(a(\cot A/2) : :\)

• Trilinears: \(a^{2}/(1 - \cos A) : :\)

• Trilinears: \(a(2ar - S) : :\)

• Barycentrics: \(\operatorname{area}(A'BC) : :\), where A'B'C' = 1st circumperp triangle

▸ Linear Combinations2 combinations

• X₅₅ = \(R\) X₁ + \(r\) X₃

• X₅₅ = \((R_{a}+R_{b}+R_{c})\) X₁ + \(r\) J_a + \(r\) J_b + \(r\) J_c, where J_a, J_b, J_c are excenters, and \(R_{a}\), \(R_{b}\), \(R_{c}\) are the exradii

▸ Incident Central Lines63 lines

{1, 3}, {2, 11}, {4, 12}, {5, 498}, {6, 31}, {7, 2346}, {8, 21}, {9, 200}, {10, 405}, {15, 203}, {16, 202}, {19, 25}, {20, 388}, {30, 495}, {34, 227}, {41, 220}, {43, 238}, {45, 678}, {48, 154}, {63, 518}, {64, 73}, {77, 1037}, {78, 960}, {81, 1002}, {92, 243}, {103, 109}, {104, 1000}, {108, 196}, {140, 496}, {181, 573}, {182, 613}, {183, 350}, {184, 215}, {192, 385}, {199, 1030}, {201, 774}, {204, 1033}, {219, 284}, {223, 1456}, {226, 516}, {255, 601}, {256, 983}, {329, 1005}, {376, 1056}, {386, 595}, {392, 997}, {411, 962}, {511, 611}, {515, 1012}, {519, 956}, {574, 1015}, {603, 963}, {631, 1058}, {650, 884}, {654, 926}, {748, 899}, {840, 901}, {846, 984}, {869, 893}, {1026, 1083}, {1070, 1076}, {1072, 1074}, {2195, 5452}

▸ Functional Relations55 relations

• X₅₅ = center of homothety of three triangles: tangential, intangents, and extangents. Also, X₅₅ is the pole-with-respect-to-the-circumcircle of the trilinear polar of X₁. These properties and others are given in

• X₅₅ = reflection of X_i in X_j for these (i,j): (1478, 495), (2099, 1)

• X₅₅ = isogonal conjugate of X₇

• X₅₅ = isotomic conjugate of X₆₀₆₃

• X₅₅ = complement of X₃₄₃₄

• X₅₅ = anticomplement of X₂₈₈₆

• X₅₅ = centroid of curvatures of circumcircle and excircles

• X₅₅ = circumcircle-inverse of X₁₁₅₅

• X₅₅ = antigonal conjugate of polar conjugate of X₃₇₇₆₇

• X₅₅ = X_i-Ceva conjugate of X_j for these (i,j): (1, 6), (3, 198), (7, 218), (8, 219), (9, 220), (21, 9), (59, 101), (104, 44), (260, 259), (284, 41)

• X₅₅ = cevapoint of X₄₂ and X₂₂₈ for these (i,j)

• X₅₅ = X_i-cross conjugate of X_j for these (i,j): (41, 6), (42, 33), (228, 212)

• X₅₅ = crosspoint of X_i and X_j for these (i,j): (1, 9), (3, 268), (7, 277), (8, 281), (21, 284), (59, 101)

• X₅₅ = crosssum of X_i and X_j for these (i,j): (1, 57), (2, 145), (4, 196), (11, 514), (55, 218), (56, 222), (63, 224), (65, 226), (81, 229), (177, 234), (241, 1362), (513, 1086), (905, 1364), (1361, 1465)

• X₅₅ = crossdifference of every pair of points on line X₂₄₁X₅₁₄

• X₅₅ = X_i-Hirst inverse of X_j for these (i,j): (6, 672), (43, 241)

• X₅₅ = X₁-line conjugate of X₂₄₁

• X₅₅ = X_i-beth conjugate of X_j for these (i,j): (21, 999), (55, 31), (100, 55), (200, 200), (643, 2), (1021, 1024)

• X₅₅ = insimilicenter of the intangents and extangents circles

• X₅₅ = insimilicenter of the intangents and tangential circles

• X₅₅ = exsimilicenter of then extangents and tangential circles

• X₅₅ = X₂₂-of-intouch-triangle

• X₅₅ = trilinear pole of line X₆₅₇X₆₆₃ (polar of X₃₃₁ wrt polar circle)

• X₅₅ = pole wrt polar circle of trilinear polar of X₃₃₁

• X₅₅ = X₄₈-isoconjugate (polar conjugate) of X₃₃₁

• X₅₅ = homothetic center of ABC and Mandart-incircle triangle

• X₅₅ = inverse-in-Feuerbach-hyperbola of X₁₀₀₁

• X₅₅ = inverse-in-circumconic-centered-at-X₁ of X₁₉₃₆

• X₅₅ = {X₁,X₄₀}-harmonic conjugate of X₆₅

• X₅₅ = trilinear square of X₂₅₉

• X₅₅ = Danneels point of X₁₀₀

• X₅₅ = vertex conjugate of PU(48)

• X₅₅ = vertex conjugate of foci of Mandart inellipse

• X₅₅ = excentral isotomic conjugate of X₂₉₄₂

• X₅₅ = homothetic center of the reflections of the intangents and extangents triangles in their respective Euler lines

• X₅₅ = perspector of ABC and extraversion triangle of X₅₆

• X₅₅ = trilinear product of PU(104)

• X₅₅ = barycentric product of PU(112)

• X₅₅ = bicentric sum of PU(112)

• X₅₅ = PU(112)-harmonic conjugate of X₆₅₀

• X₅₅ = perspector of ABC and unary cofactor triangle of 7th mixtilinear triangle

• X₅₅ = perspector of 4th mixtilinear triangle and unary cofactor triangle of 7th mixtilinear triangle

• X₅₅ = perspector of unary cofactor triangles of 3rd, 4th and 5th extouch triangles

• X₅₅ = {X₃₅₁₃,X₃₅₁₄}-harmonic conjugate of X₅₆

• X₅₅ = perspector of ABC and cross-triangle of ABC and extangents triangle

• X₅₅ = perspector of ABC and cross-triangle of ABC and Hutson extouch triangle

• X₅₅ = homothetic center of ABC and cross-triangle of ABC and 2nd Johnson-Yff triangle

• X₅₅ = Thomson-isogonal conjugate of X₅₆₅₇

• X₅₅ = homothetic center of midarc triangle and 2nd-circumperp-of-1st-circumperp triangle (which is also 1st-circumperp-of-2nd-circumperp triangle)

• X₅₅ = homothetic center of 2nd midarc triangle and 1st-circumperp-of-1st-circumperp triangle (which is also 2nd-circumperp-of-2nd-circumperp triangle)

• X₅₅ = Cundy-Parry Phi transform of X₉₄₂

• X₅₅ = Cundy-Parry Psi transform of X₉₄₃

• X₅₅ = X₄-of-1st-Johnson-Yff-triangle

• X₅₅ = homothetic center of anti-Hutson intouch triangle and anti-tangential midarc triangle

• X₅₅ = barycentric product of circumcircle intercepts of excircles radical circle

Notes

O. Bottema and J. T. Groenman, “De gemene raaklijnen van de vier raakcirkels van een driehoek, twee aan twee,” Nieuw Tijdschrift voor Wiskunde 67 (1979-80) 177-182.

Let A', B', C' be the second points of intersection of the angle bisectors of triangle ABC with its incircle. Let O_a be the circle externally tangent to the incircle at A', and internally tangent to the circumcircle; define O_b and O_c cyclically. Then X₅₅ is the radical center of circles O_a, O_b, O_c. Let A″ be the touchpoint of O_a and the circumcircle, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₅₅. Let B_a, C_a be the intersections of lines CA, AB, respectively, and the antiparallel to BC through a point P. Define C_b, A_b, A_c, B_c cyclically. Triangles AB_aC_a, A_bBC_b, A_cB_cC are congruent only when P = X₅₅ or one of its 3 extraversions. Let A*B*C* be the incentral triangle. Let L_a be the reflection of line BC in line AA*, and define L_b and L_c cyclically. Let A’’’ = L_b∩L_c, and define B’’’ and C’’’. The lines A*A’’’, B*B’’’, C*C’’’ concur in X₅₅. (Randy Hutson, November 18, 2015)

Let N_aN_bN_c be the extouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line N_bN_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₅₅. (Randy Hutson, July 31 2018)

Let (O_a) be the circumcircle of BCX₁. Let P_a be the perspector of (O_a). Let L_a be the polar of P_a wrt (O_a). Define L_b and L_c cyclically. The lines L_a, L_b, L_c concur in X₅₅. (Randy Hutson, July 31 2018)

X₅₅ is the {X₁,X₃}-harmonic conjugate of X₅₆.