X(37) = CROSSPOINT OF INCENTER AND CENTROID¶
Trilinears
\(b + c : c + a : a + b\)
\(ar - S : br - S : cr - S\)
\(semi-major axis of A-Soddy ellipse : :\)
Barycentrics
\(a(b + c) : b(c + a) : c(a + b)\)
Notes
Let A’B’C’ be the cevian triangle of X(1). Let A″ be the centroid of triangle AB’C’, and define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X(37). (Eric Danneels, Hyacinthos 7892, 9/13/03)
A simple construction of X(37) 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, X(2). The crosspoint of X(2) 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,U) = b/w+c/v : c/u+a/w : a/v+b/u, assuming that uvw is nonzero. In particular, if U=:ref:X(1) <X(1)>, then the crosspoint is X(37). (Seiichi Kirikami, July 10, 2011)
X(37) = perspector of ABC and the medial triangle of the incentral triangle of ABC. (Randy Hutson, August 23, 2011)
X(37) = center of the Hofstadter ellipse E(1/2); see X(359). This is the incentral inellipse, which is the trilinear square of the antiorthic axis. (Randy Hutson, August 9, 2014)
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 X(37). (Randy Hutson, April 9, 2016)
Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(ra^2 + s^2), where ra is the A-exradius). Let La be the radical axis of the circumcircle and Oa. Let A’ = Lb∩Lc, B’ = Lc∩La, C’ = La∩Lb. The lines AA’, BB’, CC’ concur in X(37). (Randy Hutson, April 9, 2016)
If you have The Geometer’s Sketchpad, you can view X(37). If you have GeoGebra, you can view X(37).
X(37) lies on these 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
X(37) is the {X(1),:ref:X(9) <X(9)>}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(37), click Tables at the top of this page.
X(37) = midpoint of X(i) and X(j) for these (i,j): (75,192), (190,335)
X(37) = isogonal conjugate of X(81)
X(37) = isotomic conjugate of X(274)
X(37) = complement of X(75)
X(37) = complementary conjugate of X(2887)
X(37) = anticomplement of X(3739)
X(37) = circumcircle-inverse of X(32758)
X(37) = 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)
X(37) = cevapoint of X(213) and X(228)
X(37) = X(i)-cross conjugate of X(j) for these (i,j): (42,65), (228,72)
X(37) = crosspoint of X(i) and X(j) for these (i,j): (1,2), (9,281), (10,226)
X(37) = X(1)-line conjugate of X(238)
X(37) = crosssum of X(i) and X(j) for these (i,j): (1,6), (57,222), (58,284), (1333,1437)
X(37) = crossdifference of every pair of points on line X(36)X(238)
X(37) = X(10)-Hirst inverse of X(740)
X(37) = X(1)-aleph conjugate of X(1051)
X(37) = X(i)-beth conjugate of X(j) for these (i,j): (9,37), (644,37), (645,894), (646,37), (1018,37)
X(37) = midpoint of PU(i), for these i: 6, 52, 53
X(37) = bicentric sum of PU(i), forthese i: 6, 52, 53
X(37) = trilinear product of PU(32)
X(37) = center of circumconic that is locus of trilinear poles of lines passing through X(10)
X(37) = perspector of circumconic centered at X(10)
X(37) = trilinear pole of line X(512)X(661) (polar of X(286) wrt polar circle)
X(37) = trilinear pole wrt medial triangle of Gergonne line
X(37) = pole wrt polar circle of trilinear polar of X(286) (line X(693)X(905))
X(37) = X(48)-isoconjugate (polar conjugate) of X(286)
X(37) = {X(6),:ref:X(9) <X(9)>}-harmonic conjugate of X(44)
X(37) = X(160)-of-intouch triangle
X(37) = perpector of ABC and n(Medial)*n(Incentral) triangle
X(37) = homothetic center of medial triangle and inverse of n(Medial)*n(Incentral) triangle
X(37) = perspector of incentral triangle and tangential triangle, wrt incentral triangle, of circumconic of incentral triangle centered at X(1) (the bicevian conic of X(1) and X(57))
X(37) = inverse-in-circumconic-centered-at-X(9) of X(1757)
X(37) = complement wrt incentral triangle of X(2667)
X(37) = perspector of ABC and unary cofactor triangle of 2nd circumperp triangle
X(37) = perspector of medial triangle and Gergonne line extraversion triangle
X(37) = trilinear pole, wrt Gergonne line extraversion triangle, of Gergonne line
X(37) = perspector of ABC and cross-triangle of Gemini triangles 3 and 4
X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 3
X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 4
X(37) = perspector of ABC and cross-triangle of ABC and Gemini triangle 16
X(37) = center of the {ABC, Gemini 17}-circumconic
X(37) = perspector of ABC and unary cofactor triangle of Gemini triangle 23
X(37) = incentral-to-ABC barycentric image of X(37)