X₂₈₁

X₃₇-CROSS CONJUGATE OF X₉

X₂₈₁ = isogonal conjugate of X₂₂₂

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

X₂₈₁ (X₃₇-CROSS CONJUGATE OF X₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 9}, {8, 29}

• Related Centers: X₃₇ (CROSSPOINT OF INCENTER AND CENTROID), X₉ (MITTENPUNKT), X₄ (ORTHOCENTER), X₈ (NAGEL POINT), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER)

Trilinears: \(\sec A \cot A/2 : \sec B \cot B/2 : \sec C \cot C/2\)

Barycentrics: \(\tan A \cot A/2 : \tan B \cot B/2 : \tan C \cot C/2\)

Search Value (6, 9, 13): \(5.042882205225\)

▸ Alternate Coordinates5 forms

• Trilinears: \(\csc A + 2 \csc 2A : \csc B + 2 \csc 2B : \csc C + 2 \csc 2C\)

• Trilinears: \((1 + \sec A)/a : (1 + \sec B)/b : (1 + \sec C)/c\)

• Trilinears: \((b + c - a) \sec A : :\)

• Barycentrics: \(1 + \sec A : 1 + \sec B : 1 + \sec C\)

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

▸ Incident Central Lines23 lines

{1, 282}, {2, 92}, {4, 9}, {6, 1146}, {7, 653}, {8, 29}, {28, 958}, {33, 200}, {37, 158}, {45, 53}, {48, 944}, {100, 1013}, {189, 222}, {196, 226}, {220, 594}, {240, 984}, {264, 344}, {268, 1012}, {318, 346}, {380, 950}, {451, 1068}, {515, 610}, {612, 1096}

▸ Functional Relations14 relations

• X₂₈₁ = isogonal conjugate of X₂₂₂

• X₂₈₁ = isotomic conjugate of X₃₄₈

• X₂₈₁ = complement of X₃₄₇

• X₂₈₁ = X_i-Ceva conjugate of X_j for these (i,j): (29, 33), (92, 4)

• X₂₈₁ = X_i-cross conjugate of X_j for these (i,j): (33, 4), (37, 9), (55, 8)

• X₂₈₁ = crosspoint of X_i and X_j for these (i,j): (2, 280), (92, 318)

• X₂₈₁ = crosssum of X_i and X_j for these (i,j): (6, 221), (48, 603), (73, 1409), (652, 1364)

• X₂₈₁ = {X₂,X₉₂}-harmonic conjugate of X₂₇₈

• X₂₈₁ = pole, wrt polar circle, of Gergonne line

• X₂₈₁ = X₄₈-isoconjugate (polar conjugate) of X₇

• X₂₈₁ = crossdifference of every pair of points on line X₁₄₅₉X₁₉₄₆

• X₂₈₁ = trilinear pole of line X₃₀₆₄X₃₇₀₀ (the polar of X₇ wrt polar circle)

• X₂₈₁ = perspector of Gemini triangle 37 and cross-triangle of ABC and Gemini triangle 37

• X₂₈₁ = trilinear pole of line X₃₀₆₄X₃₇₀₀ (the perspectrix of ABC and Gemini triangle 38)

Notes

X₂₈₁: Let T_aT_bT_c be the intouch triangle. Let A″ be the pole, wrt the polar circle, of line T_bT_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₈₁. (Randy Hutson, September 14, 2016)

Let A'B'C' be the 3rd extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The triangle A'B'C' is homothetic to ABC at X₂₈₁. (Randy Hutson, September 14, 2016)

Let A'B'C' be the 4th extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₈₁. (Randy Hutson, September 14, 2016)

Let A'B'C' be the 5th extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₈₁. (Randy Hutson, September 14, 2016)

Let E_a be the ellipse passing through A, and with foci at B and C (the A-Soddy ellipse). Let L_a be the polar of X₃ with respect to E_a. Define L_b and L_c cyclically. Let A' = L_b∩L_c, B' = L_c∩L_a, C' = L_a∩L_b. Triangle A'B'C' is homothetic to ABC, and the center of homothety is X₂₈₁. (Randy Hutson, September 14, 2016)