X₃₃

PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES

X₃₃ is the {X₁,X₄}-harmonic conjugate of X₃₄. For a list of other harmonic conjugates, click Tables at the top of this page.

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

X₃₃ (PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {19, 25}

• Related Centers: X₇₇ (ISOGONAL CONJUGATE OF X₃₃), X₁ (INCENTER), X₄ (ORTHOCENTER), X₁₉ (CLAWSON POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES)

Trilinears: \(1 + \sec A : 1 + \sec B : 1 + \sec C\)

Barycentrics: \(\sin A + \tan A : \sin B + \tan B : \sin C + \tan C\)

Search Value (6, 9, 13): \(5.246983589481\)

▸ Alternate Coordinates4 forms

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

• Trilinears: \((b + c - a)/(b^{2} + c^{2} - a^{2}) : :\)

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

• Barycentrics: \(\tan A \cos^{2}(A/2) : :\)

▸ Linear Combinations1 combination

• X₃₃ = \((r + 2R - s)(r + 2R + s)\) X₁ - \(6rR\) X₂ + \(4rR\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines32 lines

{1, 4}, {2, 1040}, {5, 1062}, {6, 204}, {7, 1041}, {8, 1039}, {9, 212}, {10, 406}, {11, 427}, {12, 235}, {19, 25}, {20, 1038}, {24, 35}, {28, 975}, {29, 78}, {30, 1060}, {36, 378}, {40, 201}, {42, 393}, {47, 90}, {56, 963}, {57, 103}, {63, 1013}, {64, 65}, {79, 1063}, {80, 1061}, {84, 603}, {112, 609}, {200, 281}, {210, 220}, {222, 971}, {264, 350}

▸ Functional Relations14 relations

• X₃₃ = isogonal conjugate of X₇₇

• X₃₃ = isotomic conjugate of X₇₁₈₂

• X₃₃ = anticomplement of X₃₄₈₂₂

• X₃₃ = trilinear pole of line X₆₅₇X₄₀₄₁ (the polar of X₈₅ wrt polar circle)

• X₃₃ = pole wrt polar circle of trilinear polar of X₈₅ (line X₅₂₂X₆₉₃)

• X₃₃ = polar conjugate of X₈₅

• X₃₃ = perspector of ABC and extraversion triangle of X₃₄

• X₃₃ = X_i-Ceva conjugate of X_j for these (i,j): (4, 19), (29, 281), (318, 9)

• X₃₃ = X_i-cross conjugate of X_j for these (i,j): (41, 9), (42, 55)

• X₃₃ = crosspoint of X_i and X_j for these (i,j): (1, 282), (4, 281)

• X₃₃ = crosssum of X_i and X_j for these (i,j): (1, 223), (3, 222), (57, 1394), (73, 1214)

• X₃₃ = crossdifference of every pair of points on line X₆₅₂X₉₀₅

• X₃₃ = X₃₃-beth conjugate of X₂₅

• X₃₃ = homothetic center of anti-excenters-incenter reflections triangle and anti-tangential midarc triangle

Notes

Let L_A be the reflection of line BC in the internal angle bisector of angle A, and define L_B and L_C cyclically. Let DEF be the triangle formed by L_A, L_B, L_C. Then DEF (the intangents triangle) is homothetic to the orthic triangle, and the homothetic center is X₃₃. (Randy Hutson, 9/23/2011)