X₃₅₈₉

ISOGONAL CONJUGATE OF X₃₁₀₈

P' = 2u + v + w : u + 2v + w : u + v +2w.

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

X₃₅₈₉ (ISOGONAL CONJUGATE OF X₃₁₀₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Related Centers: X₃₁₀₈ (DC(X₈₃))

Trilinears: \(bc(2a^{2} + b^{2} + c^{2}) : :\)

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

Search Value (6, 9, 13): \(2.220253718133\)

▸ Linear Combinations1 combination

• X₃₅₈₉ = \(3\) X₂ + X₆

X₃₅₈₉ lies on these lines:

▸ Functional Relations22 relations

• X₃₅₈₉ = midpoint of X_i and X_j for these (i,j): (2, 597), (5, 182), (6, 141), (10, 1386), (625, 2030)

• X₃₅₈₉ = isogonal conjugate of X₃₁₀₈

• X₃₅₈₉ = isotomic conjugate of X₁₀₁₅₉

• X₃₅₈₉ = complement of X₁₄₁

• X₃₅₈₉ = complement of complement of X₆

• X₃₅₈₉ = anticomplement of X₃₄₅₇₃

• X₃₅₈₉ = polar conjugate of isogonal conjugate of X₂₂₃₅₂

• X₃₅₈₉ = polar conjugate of isotomic conjugate of X₇₇₆₇

• X₃₅₈₉ = complementary conjugate of X₂₁₂₄₈

• X₃₅₈₉ = crosssum of X₆ and X₃₉

• X₃₅₈₉ = {X₂,X₆}-harmonic conjugate of X₁₄₁

• X₃₅₈₉ = centroid of {A, B, C,X₆}

• X₃₅₈₉ = centroid of PU(11)PU(45)

• X₃₅₈₉ = Kosnita(X₆,X₂) point

• X₃₅₈₉ = crosspoint of X₂ and X₈₃

• X₃₅₈₉ = X₆₂₀ of 1st Brocard triangle

• X₃₅₈₉ = antipode of X₁₄₁ in conic {{X₁₃,X₁₄,X₁₅,X₁₆,X₁₄₁}}

• X₃₅₈₉ = complement of isotomic conjugate of cevapoint of X₂ and X₆

• X₃₅₈₉ = polar conjugate of isogonal conjugate of X₂₂₃₅₂

• X₃₅₈₉ = perspector of the medial triangle and the tangential triangle, wrt the symmedial triangle, of the bicevian conic of X₂ and X₆

• X₃₅₈₉ = crossdifference of every pair of points on line X₅₂₃X₂₀₇₆

• X₃₅₈₉ = X₉-of-submedial-triangle if ABC is acute

Notes

In 2003, Peter Moses gave a general construction as follows: Suppose that P =\(u : v : w\) (barycentrics). The centroids of the triangles BCP, CAP, ABP form a triangle homothetic to ABC, with ratio -1/3 and center

In 2010, Seiichi Kirikami gave another construction for P': let

D = AP∩BC, E = BP∩CA, F = CP∩AB.

Then the Newton lines of the quadrilaterals PEAF, PFBD, PDCE concur in P'. (The Newton line of a quadrilateral is the line of the midpoints of the two diagonals of the quadrilateral.)

X₃₅₈₉ is the Moses-Kirikami image of the symmedian point. In the following list (from P. Moses, Aug. 23, 2010), the appearance of I, J means that X_j is the Moses-Kirikami image of X_i.

1,1125 2,6 3,140 4,5 6,3589 7,142 8,10 20,3 23,468 69,141 99,620 100,3035 144,9 145,1 146,113 147,114 148,115 149,11 150,116 151,117 152,118 153,119 192,37 193,6 194,39 239,3008 315,626 316,625 329,3452 376,549 381,547 382,546 385,230 390,1001 410,441 487,642 488,641 550,3530 616,618 617,619 621,623 622,624 627,629 628,630 631,632 633,635 634,636 637,639 638,640 944,1385 962,946 1278,75 1320,1387 1330,3454 1370,1368 1654,1213 1657,548 1916,2023 1992,597 2475,442 2888,1209 2895,1211 3091,1656 3146,4 3151,440 3153,2072 3164,216 3177,1212 3180,396 3181,395 3241,551 3434,2886 3436,1329 3448,125 3522,631 3523,3526 3529,550 3543,381

Early in 2012, Seiichi Kirikami found a simple relation between the tetrahedron and the Moses-Kirikami image. Using 3-dimensional cartesian coordinates, suppose that the vertices of triangle ABC are placed in the xy-plane:

A = (\(x_{1}\), \(y_{1}\), 0), B = (\(x_{2}\), \(y_{2}\), 0), C = (\(x_{3}\), \(y_{3}\), 0).

Let P = (\(x_{4}\), \(y_{4}\), \(z_{4})\) be an arbitrary point not in the plane of ABC. Then the centroid of the tetrahedron ABCP is the point

Q = ( (\(x_{1}\) + \(x_{2}\) + \(x_{3}\) + \(x_{4})\)/4, (\(y_{1}\) + \(y_{2}\) + \(y_{3}\) + \(y_{4})\)/4, \(z_{4}\)/4 ).

The orthogonal projections of P and Q onto the xy plane coincide with the Moses-Kirikami images of X₆ and X₃₅₈₉. That is, X₆ has coordinates (\(x_{4}\), \(y_{4}\), 0), and X₃₅₈₉ has coordinates ( (\(x_{1}\) + \(x_{2}\) + \(x_{3}\) + \(x_{4})\)/4, (\(y_{1}\) + \(y_{2}\) + \(y_{3}\) + \(y_{4})\)/4, 0 ).

Let A' be the midpoint of segment AX₆, and define B' and C' cyclically. Then A'B'C' is homothetic to the medial triangle, and the homothetic center is X₃₅₈₉. (Randy Hutson, December 26, 2015)

Let E be the bicevian conic of X₂ and X₆; i.e., the ellipse that passes through the vertices of the medial and symmedial triangles. Then X₃₅₈₉ is the center of E. This ellipse is also the locus of centers of circumconics passing through X₆. Let L be a line through X₂, and let P and P' be the points of intersection of L and the circumcircle. Let V be the locus of the crosssum of P and P'. The locus of V generated by L is E. (Randy Hutson, December 26, 2015)

Let A' be the reflection of X₆ in line BC. Let O_a be the circle with center A' and tangent to BC. Define O_b and O_c cyclically. The radical center of O_a, O_b, O_c is X₃₅₈₉. (Randy Hutson, December 26, 2015)

2,6 5,182 10,1386 11,2330 12,1428 23,2916 39,698 51,3313 53,458 83,316 140,143 216,441 239,594 264,1990 373,468 397,622 398,621 427,1974 498,613 499,611 518,1125 542,547 549,3098 575,3564 576,632 625,2030 631,1350 742,3008 894,1086 1351,3526 1352,1656 1506,1692 1698,3416 2450,3566