X₃₇

CROSSPOINT OF INCENTER AND CENTROID

X₃₇ is the crosspoint of incenter and centroid.

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

X₃₇ (CROSSPOINT OF INCENTER AND CENTROID) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {19, 25}

• Related Centers: X₁ (INCENTER), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₁₉ (CLAWSON POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES)

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

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

Tripolars: \((pending)\)

Search Value (6, 9, 13): \(2.090823377160\)

▸ Alternate Coordinates2 forms

• Trilinears: \(ar - S : br - S : cr - S\)

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

▸ Linear Combinations1 combination

• X₃₇ = \((r^{2} + 2rR - s^{2})\) X₁ - \(6rR\) X₂ - \(2r^{2}\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines39 lines

{1, 6}, {2, 75}, {3, 975}, {7, 241}, {8, 941}, {10, 594}, {12, 225}, {19, 25}, {21, 172}, {35, 267}, {38, 354}, {39, 596}, {41, 584}, {48, 205}, {63, 940}, {65, 71}, {73, 836}, {78, 965}, {82, 251}, {86, 190}, {91, 498}, {100, 111}, {101, 284}, {141, 742}, {142, 1086}, {145, 391}, {158, 281}, {171, 846}, {226, 440}, {256, 694}, {347, 948}, {513, 876}, {517, 573}, {537, 551}, {579, 942}, {626, 746}, {665, 900}, {971, 991}, {1953, 2183}

▸ Functional Relations45 relations

• X₃₇ = perspector of ABC and the medial triangle of the incentral triangle of ABC. (Randy Hutson, August 23, 2011)

• X₃₇ = center of the Hofstadter ellipse E(1/2); see X₃₅₉. This is the incentral inellipse, which is the trilinear square of the antiorthic axis. (Randy Hutson, August 9, 2014)

• X₃₇ = midpoint of X_i and X_j for these (i,j): (75, 192), (190, 335)

• X₃₇ = isogonal conjugate of X₈₁

• X₃₇ = isotomic conjugate of X₂₇₄

• X₃₇ = complement of X₇₅

• X₃₇ = complementary conjugate of X₂₈₈₇

• X₃₇ = anticomplement of X₃₇₃₉

• X₃₇ = circumcircle-inverse of X₃₂₇₅₈

• X₃₇ = X_i-Ceva conjugate of X_j for these (i,j): (1, 42), (2, 10), (4, 209), (9, 71), (10, 210), (190, 513), (226, 65), (321, 72), (335, 518)

• X₃₇ = cevapoint of X₂₁₃ and X₂₂₈

• X₃₇ = X_i-cross conjugate of X_j for these (i,j): (42, 65), (228, 72)

• X₃₇ = crosspoint of X_i and X_j for these (i,j): (1, 2), (9, 281), (10, 226)

• X₃₇ = X₁-line conjugate of X₂₃₈

• X₃₇ = crosssum of X_i and X_j for these (i,j): (1, 6), (57, 222), (58, 284), (1333, 1437)

• X₃₇ = crossdifference of every pair of points on line X₃₆X₂₃₈

• X₃₇ = X₁₀-Hirst inverse of X₇₄₀

• X₃₇ = X₁-aleph conjugate of X₁₀₅₁

• X₃₇ = X_i-beth conjugate of X_j for these (i,j): (9, 37), (644, 37), (645, 894), (646, 37), (1018, 37)

• X₃₇ = midpoint of PU(i), for these i: 6, 52, 53

• X₃₇ = bicentric sum of PU(i), forthese i: 6, 52, 53

• X₃₇ = trilinear product of PU(32)

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

• X₃₇ = perspector of circumconic centered at X₁₀

• X₃₇ = trilinear pole of line X₅₁₂X₆₆₁ (polar of X₂₈₆ wrt polar circle)

• X₃₇ = trilinear pole wrt medial triangle of Gergonne line

• X₃₇ = pole wrt polar circle of trilinear polar of X₂₈₆ (line X₆₉₃X₉₀₅)

• X₃₇ = X₄₈-isoconjugate (polar conjugate) of X₂₈₆

• X₃₇ = {X₆,X₉}-harmonic conjugate of X₄₄

• X₃₇ = X₁₆₀-of-intouch triangle

• X₃₇ = perpector of ABC and n(Medial)*n(Incentral) triangle

• X₃₇ = homothetic center of medial triangle and inverse of n(Medial)*n(Incentral) triangle

• X₃₇ = perspector of incentral triangle and tangential triangle, wrt incentral triangle, of circumconic of incentral triangle centered at X₁ (the bicevian conic of X₁ and X₅₇)

• X₃₇ = inverse-in-circumconic-centered-at-X₉ of X₁₇₅₇

• X₃₇ = complement wrt incentral triangle of X₂₆₆₇

• X₃₇ = perspector of ABC and unary cofactor triangle of 2nd circumperp triangle

• X₃₇ = perspector of medial triangle and Gergonne line extraversion triangle

• X₃₇ = trilinear pole, wrt Gergonne line extraversion triangle, of Gergonne line

• X₃₇ = perspector of ABC and cross-triangle of Gemini triangles 3 and 4

• X₃₇ = perspector of ABC and cross-triangle of ABC and Gemini triangle 3

• X₃₇ = perspector of ABC and cross-triangle of ABC and Gemini triangle 4

• X₃₇ = perspector of ABC and cross-triangle of ABC and Gemini triangle 16

• X₃₇ = center of the {ABC, Gemini 17}-circumconic

• X₃₇ = perspector of ABC and unary cofactor triangle of Gemini triangle 23

• X₃₇ = incentral-to-ABC barycentric image of X₃₇

Notes

Let I_aI_bI_c be the cevian triangle of X₁. Let A″ be the centroid of triangle AI_bI_c, and define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X₃₇. (Eric Danneels, Hyacinthos 7892, 9/13/03)

A simple construction of X₃₇ as a crosspoint can be generalized as follows: let DEF be the medial triangle of ABC and let A'B'C' be the cevian triangle of a point U other than the centroid, X₂. The crosspoint of X₂ and U is then the point of concurrence of lines LD,ME,NF, where L,M,N are the respective midpoints of AA', BB', CC'. If U=\(u : v : w\) (trilinears), then crosspoint(X₂,U) = b/w+c/v : c/u+a/w : a/v+b/u, assuming that uvw is nonzero. In particular, if U=X₁, then the crosspoint is X₃₇. (Seiichi Kirikami, July 10, 2011)

Let A' be the trilinear pole of the tangent to the Apollonius circle where it meets the A-excircle, and define B' and C' cyclically. The triangle A'B'C' is homothetic to ABC at X₃₇. (Randy Hutson, April 9, 2016)

Let O_a be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius \(\sqrt{r_{a}^{2} + s^{2}}\), where \(r_{a}\) is the A-exradius). Let L_a be the radical axis of the circumcircle and O_a. Let A' = L_b∩L_c, B' = L_c∩L_a, C' = L_a∩L_b. The lines AA', BB', CC' concur in X₃₇. (Randy Hutson, April 9, 2016)

X₃₇ is the {X₁,X₉}-harmonic conjugate of X₆.