X₇

GERGONNE POINT

Let T_a, T_b, T_c denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines AT_a, BT_b, CT_c concur in X₇.

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

X₇ (GERGONNE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

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

• Concurrence Lines: A–T_a, B–T_b, C–T_c concurring at X₇

• Related Centers: X₁ (INCENTER)

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

Barycentrics: \(1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)\)

Tripolars: \((b+c-a) \sqrt{a (a-b-c) (a^{2}+a b-2 b^{2}+a c+4 b c-2 c^{2})} : :\)

Search Value (6, 9, 13): \(0.744160979006\)

▸ Alternate Coordinates7 forms

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

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

• Trilinears: \((bc - S_{A})/a : (ca - S_{B})/b : (ab - S_{C})/c\)

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

• Barycentrics: \(\tan A/2 : \tan B/2 : \tan C/2\)

• Barycentrics: \(bc - S_{A} : ca - S_{B} : ab - S_{C}\)

• Barycentrics: \(r_{a} : r_{b} : r_{c}\), where \(r_{a}\), \(r_{b}\), \(r_{c}\) are the exradii

▸ Linear Combinations2 combinations

• X₇ = \((2r + 4R)\) X₁ + \(3r\) X₂ - \(4r\) X₃

• X₇ = A / \(R_{a}\) + B / \(R_{b}\) + C / \(R_{c}\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) = radii of Soddy circles

X₇ lies on the Lucas cubic.

Related Collection 1 Topic Guides

T_2: Isotomic Conjugate • T_11: The Soddy Line • T_12: The Fuhrmann & Spieker Central Lines • T_14: The Incircle & Excircles • T_22: Intouch (Contact) & Extouch Triangles

▸ Incident Central Lines78 lines

{1, 20}, {2, 9}, {3, 943}, {4, 273}, {6, 294}, {8, 65}, {11, 658}, {12, 1268}, {21, 56}, {27, 81}, {37, 241}, {33, 1041}, {34, 1039}, {55, 2346}, {58, 272}, {59, 1275}, {72, 443}, {73, 1246}, {76, 1479}, {80, 150}, {92, 189}, {100, 1004}, {104, 934}, {108, 1013}, {109, 675}, {145, 1266}, {171, 983}, {174, 234}, {177, 555}, {190, 344}, {192, 335}, {193, 239}, {218, 277}, {220, 1223}, {225, 969}, {238, 1471}, {253, 280}, {256, 982}, {274, 959}, {281, 653}, {286, 331}, {310, 314}, {330, 1432}, {349, 1269}, {354, 479}, {404, 1259}, {452, 1467}, {464, 1214}, {480, 1376}, {492, 1267}, {513, 885}, {517, 1000}, {528, 664}, {554, 1082}, {594, 599}, {604, 1429}, {757, 1414}, {840, 927}, {857, 1901}, {870, 1431}, {940, 1407}, {941, 1427}, {944, 1389}, {952, 1159}, {986, 1254}, {987, 1106}, {1002, 1362}, {1020, 1765}, {1061, 1870}, {1210, 3091}, {1354, 1367}, {1365, 1366}, {1386, 1456}, {1419, 1449}, {1435, 1848}, {1486, 1602}, {1617, 1621}, {2475, 2893}

▸ Functional Relations64 relations

• X₇ = reflection of X_i in X_j for these (i,j): (9, 142), (144, 9), (390, 1), (673, 1086), (1156, 11)

• X₇ = isogonal conjugate of X₅₅

• X₇ = isotomic conjugate of X₈

• X₇ = cyclocevian conjugate of X₇

• X₇ = circumcircle-inverse of (32624)

• X₇ = incircle-inverse of (1323)

• X₇ = complement of X₁₄₄

• X₇ = anticomplement of X₉

• X₇ = complementary conjugate of X₂₈₈₄

• X₇ = anticomplementary conjugate of X₃₂₉

• X₇ = X_i-Ceva conjugate of X_j for these (i,j): (75, 347), (85, 2), (86, 77), (286, 273), (331, 278)

• X₇ = cevapoint 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)

• X₇ = X_i-cross conjugate of X_j for these (i,j): (1, 2), (4, 189), (11, 514), (55, 277), (56, 278), (57, 279), (65, 57), (177, 174), (222, 348), (226, 85), (354, 1), (497, 8), (517, 88)

• X₇ = crosspoint of X_i and X_j for these (i,j): (75, 309), (86, 286)

• X₇ = crosssum of X_i and X_j for these (i,j): (41, 1253), (42, 228)

• X₇ = crossdifference of every pair of points on line X₆₅₇X₆₆₃

• X₇ = X₅₇-Hirst inverse of X₁₄₄₇

• X₇ = insimilicenter of inner and outer Soddy circles; the exsimilicenter is X₁

• X₇ = X_i-beth conjugate of X_j for these (i,j): (7, 269), (21, 991), (75, 75), (86, 7), (99, 7), (190, 7), (290, 7), (314, 69), (645, 344), (648, 7), (662, 6), (664, 7), (666, 7), (668, 7), (670, 7), (671, 7), (811, 264), (886, 7), (889, 7), (892, 7), (903, 7)

• X₇ = vertex conjugate of foci of inellipse that is isotomic conjugate of isogonal conjugate of incircle (centered at X₂₈₈₆)

• X₇ = trilinear product of vertices of Hutson-extouch triangle

• X₇ = orthocenter of X₄X₈X₈₈₅

• X₇ = trilinear cube of X₅₀₆

• X₇ = barycentric product of PU(47)

• X₇ = trilinear product of PU(94)

• X₇ = vertex conjugate of PU(95)

• X₇ = bicentric sum of PU(120)

• X₇ = perspector of ABC and its intouch triangle.

• X₇ = perspector of ABC and the reflection in X₅₇ of the pedal triangle of X₅₇

• X₇ = perspector of AC-incircle

• X₇ = X₆-of-extraversion triangle-of-X₈

• X₇ = X₆-of-intouch-triangle; X₇ is the only point X inside ABC Such that X(ABC) = X(A'B'C'), where A'B'C' = cevian triangle of X

• X₇ = {X₂,X₆₃}-harmonic conjugate of X₅₂₇₃

• X₇ = {X₉,X₅₇}-harmonic conjugate of X₁₄₄₅

• X₇ = {X₁₃₇₁,X₁₃₇₂}-harmonic conjugate of X₁

• X₇ = {X₁₃₇₃,X₁₃₇₄}-harmonic conjugate of X₁

• X₇ = trilinear pole of Gergonne line

• X₇ = trilinear pole, wrt intouch triangle, of Gergonne line

• X₇ = pole of Gergonne line wrt incircle

• X₇ = pole wrt polar circle of trilinear polar of X₂₈₁ (line X₃₀₆₄X₃₇₀₀)

• X₇ = X₄₈-isoconjugate (polar conjugate) of X₂₈₁

• X₇ = X₆-isoconjugate of X₉

• X₇ = X₇₅-isoconjugate of X₂₁₇₅

• X₇ = X₁₁₀₁-isoconjugate of X₄₀₉₂

• X₇ = perspector of circumconic centered at X₃₁₆₀

• X₇ = center of circumconic that is locus of trilinear poles of lines passing through X₃₁₆₀

• X₇ = X₂-Ceva conjugate of X₃₁₆₀

• X₇ = antigonal image of X₁₁₅₆

• X₇ = homothetic center of intouch triangle and anticomplement of the excentral triangle

• X₇ = X₆-of-intouch-triangle; X₇ is the only point X inside ABC such that X(ABC) = X(A'B'C'), where A'B'C' = cevian triangle of X

• X₇ = perspector of ABC and cross-triangle of inner and outer Soddy triangles

• X₇ = perspector of excentral triangle and cross-triangle of ABC and Honsberger triangle

• X₇ = perspector of inverse-in-excircles triangle and cross-triangle of ABC and inverse-in-excircles triangle

• X₇ = perspector of inverse-in-incircle triangle and cross-triangle of ABC and inverse-in-incircle triangle

• X₇ = X₁₈₄₃-of-excentral-triangle

• X₇ = Cundy-Parry Phi transform of X₉₄₃

• X₇ = Cundy-Parry Psi transform of X₉₄₂

• X₇ = {X₁,X₁₇₄₂}-harmonic conjugate of X₂₂₉₃

• X₇ = barycentric square of X₅₀₈

• X₇ = perspector of ABC and cross-triangle of ABC and Gemini triangle 40

• X₇ = barycentric product of vertices of Gemini triangle 40

• X₇ = excentral-to-intouch similarity image of X₉

• X₇ = circumconic-centered-at-X₉-inverse of X₃₇₇₈₇

• X₇ = endo-homothetic center of Ehrmann vertex-triangle and Ehrmann mid-triangle; the homothetic center is X₃₈₁₈

Notes

An exarc circle is a circle tangent to two sides of a triangle ABC and externally tangent to the circumcircle of ABC. The point whose distance to the sides BC, CA, AB are proportional to the respective radii of the exarc circles is X₂₇₉. The point having distances to the sides proportional to the radii of the inarc circles is X₇. See Martin Lukarevski, “Exarc radii and the Finsler-Hadwiger inequality”, The Mathematical Gazette 106, issue 565, March 2022, pp. 138-143.

X₇ is the {X₆₉,X₇₅}-harmonic conjugate of X₈.