X₁₁₂₅

COMPLEMENT OF X₁₀

X₁₁₂₅ is the complement of X₁₀.

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

X₁₁₂₅ (COMPLEMENT OF X₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 10}

• Related Centers: X₂ (CENTROID), X₁₀ (SPIEKER CENTER)

Trilinears: \((2a+b+c)/a : (a+2b+c)/b : (a+b+2c)/c\)

Barycentrics: \(2a+b+c : a+2b+c : a+b+2c\)

Search Value (6, 9, 13): \(2.394603721731\)

▸ Alternate Coordinates1 form

• Trilinears: \(r + 2 R \sin B \sin C : :\)

▸ Linear Combinations1 combination

• X₁₁₂₅ = X₁ + \(3\) X₂ = \(3\) X₁ + X₈ = \(3\) X₂ - X₁₀ = X₈ - \(3\) X₁₀

▸ Incident Central Lines27 lines

{1, 2}, {3, 142}, {5, 515}, {11, 214}, {21, 36}, {33, 475}, {34, 406}, {35, 404}, {37, 39}, {40, 631}, {55, 474}, {56, 226}, {58, 86}, {65, 392}, {72, 354}, {105, 831}, {114, 116}, {140, 517}, {165, 962}, {171, 595}, {274, 350}, {409, 759}, {443, 497}, {749, 984}, {758, 942}, {958, 999}, {1015, 1107}

▸ Functional Relations25 relations

• X₁₁₂₅ = midpoint of X_i and X_j for these (i,j): (1, 10), (2, 551), (3, 946), (5, 1385), (8, 3244), (11, 214), (142, 1001), (226, 993), (942, 960), (999, 3452), (1100, 3775), (4065, 4647)

• X₁₁₂₅ = isogonal conjugate of X₁₁₂₆

• X₁₁₂₅ = isotomic conjugate of X₁₂₆₈

• X₁₁₂₅ = complement of X₁₀

• X₁₁₂₅ = anticomplement of X₃₆₃₄

• X₁₁₂₅ = crosspoint of X₂ and X₈₆

• X₁₁₂₅ = crosssum of X₆ and X₄₂

• X₁₁₂₅ = perspector of circumconic centered at X₁₂₁₃

• X₁₁₂₅ = center of circumconic that is locus of trilinear poles of lines passing through X₁₂₁₃

• X₁₁₂₅ = center of bicevian conic of X₁ and X₂

• X₁₁₂₅ = Kosnita(X₁,X₂) point

• X₁₁₂₅ = X₁-Ceva conjugate of X₄₀₆₅

• X₁₁₂₅ = X₂-Ceva conjugate of X₁₂₁₃

• X₁₁₂₅ = X₂₁₄-of-X₁-Brocard triangle

• X₁₁₂₅ = complement of X₄₀₆₅ wrt incentral triangle

• X₁₁₂₅ = trilinear product of vertices of anti-Aquila triangle

• X₁₁₂₅ = X₁₀₁₁₀-of-excentral-triangle

• X₁₁₂₅ = {X₁,X₈}-harmonic conjugate of X₃₂₄₄

• X₁₁₂₅ = {X₂,X₁₀}-harmonic conjugate of X₃₆₃₄

• X₁₁₂₅ = {X₈,X₁₀}-harmonic conjugate of X₄₆₉₁

• X₁₁₂₅ = perspector of Gemini triangle 11 and cross-triangle of ABC and Gemini triangle 11

• X₁₁₂₅ = homothetic center of anticomplementary triangle and cross-triangle of Aquila and anti-Aquila triangles

• X₁₁₂₅ = trilinear pole of line X₄₉₆₉X_4976 (the perspectrix of ABC and Gemini triangle 12)

• X₁₁₂₅ = polar conjugate of isogonal conjugate of X₂₂₀₅₄

• X₁₁₂₅ = perspector of medial triangle and n(Medial)*n(Incentral) triangle

Notes

The centroid of four points A, B, C,P is the complement of the complement of P with respect to triangle ABC. As an example, X₁₁₂₅ is the centroid of {A, B, C,X₁}. (Darij Grinberg, 12/28/02)

Let A' the midpoint of segment BC and let A″ be the midpoint of segment AA'. Define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to ABC, and the center of homothety is X₁₁₂₅.

Let I be the incenter of triangle ABC and A' the centroid of triangle BCI, and define B' and C' cyclically. The triangle A'B'C' is homothetic to ABC, and the center of homothety is X₁₁₂₅.

X₁₁₂₅ is the center of the ellipse which is the locus of centers of the conics passing through A, B, C, and X₁. This ellipse is also the locus of crosssums of the intersections of the circumcircle and lines through X₁. Furthmore, this ellipse is the bicevian conic of X₁ and X₂ (i.e. the conic which passes through the vertices of the incentral and medial triangles). The ellipse passes through X₁₁, X₂₁₄, X₂₄₄, X₁₀₁₅ and the midpoints of the sides of ABC. (Randy Hutson, 8/13/2011, Hyacinthos #20179; see also #20181, by Chris van Tienhoven.)

A construction of X₁₁₂₅ is given by Antreas Hatipolakis and Angel Montesdeoca at 24185.

Let I_aI_bI_c be the incentral triangle. Let A″ be the reflection of A in I_a, and define B″ and C″ cyclically. Let A* be the trilinear pole of line B″C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₁₁₂₅. (Randy Hutson, July 21, 2017)

Let A'B'C' be the 2nd circumperp triangle. Let A″B″C″ be the triangle bounded by the Simson lines of A', B', C'. A″B″C″ is homothetic to A'B'C' at X₁₁₂₅. (Randy Hutson, July 21, 2017)

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

For an other construction of the point X₁₁₂₅ see Antreas Hatzipolakis and Peter Moses, euclid 5220.

X₁₁₂₅ is the {X₁,X₂}-harmonic conjugate of X₁₀.