X₂₇₈

ISOGONAL CONJUGATE OF X₂₁₉

X₂₇₈ is the isogonal conjugate of X₂₁₉.

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

X₂₇₈ (ISOGONAL CONJUGATE OF X₂₁₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {7, 27}

• Related Centers: X₂₁₉ (-CEVA CONJUGATE OF X₅₅), X₁ (INCENTER), X₄ (ORTHOCENTER), X₇ (GERGONNE POINT), X₂₇ (CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER)

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

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

Search Value (6, 9, 13): \(-0.270494619741\)

▸ Alternate Coordinates4 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: \(bc/[(b + c - a)(b^{2} + c^{2} - a^{2})]\)

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

▸ Incident Central Lines19 lines

{1, 4}, {2, 92}, {7, 27}, {19, 57}, {25, 105}, {28, 56}, {65, 387}, {88, 653}, {109, 917}, {219, 329}, {240, 982}, {241, 277}, {242, 459}, {274, 331}, {354, 955}, {393, 1108}, {412, 962}, {443, 1038}, {614, 1096}

▸ Functional Relations15 relations

• X₂₇₈ = isogonal conjugate of X₂₁₉

• X₂₇₈ = isotomic conjugate of X₃₄₅

• X₂₇₈ = X_i-Ceva conjugate of X_j for these (i,j): (27, 57), (92, 196), (273, 4), (331, 7)

• X₂₇₈ = cevapoint of X₁₉ and X₃₄

• X₂₇₈ = X_i-cross conjugate of X_j for these (i,j): (19, 4), (56, 7), (225, 273)

• X₂₇₈ = trilinear pole of line X₅₁₃X₁₈₃₅ (the polar of X₈ wrt polar circle, and inverse-in-polar-circle of Fuhrmann circle)

• X₂₇₈ = pole wrt polar circle of trilinear polar of X₈ (line X₅₂₂X₆₅₀)

• X₂₇₈ = X₄₈-isoconjugate (polar conjugate) of X₈

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

• X₂₇₈ = X₁₉-isoconjugate of X₁₂₅₉

• X₂₇₈ = X₉₂-isoconjugate of X₆₀₅₆

• X₂₇₈ = vertex conjugate of foci of inconic that is the polar conjugate of the isogonal conjugate of the incircle

• X₂₇₈ = perspector of Gemini triangle 38 and cross-triangle of ABC and Gemini triangle 38

• X₂₇₈ = trilinear pole of line X₅₁₃X₁₈₃₅ (the perspectrix of ABC and Gemini triangle 37)

• X₂₇₈ = Brianchon point (perspector) of inconic that is the polar conjugate of the isotomic conjugate of the incircle

Notes

Let N_aN_bN_c be the extouch triangle. Let A″ be the pole, wrt the polar circle, of line N_bN_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₇₈. (Randy Hutson, September 14, 2016)

Let A'B'C' be the 2nd extouch triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The triangle A'B'C' is homothetic to ABC at X₂₇₈. (Randy Hutson, September 14, 2016)

Let H_a be the hyperbola passing through A, and with foci at B and C. This is called the A-Soddy hyperbola in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let L_a be the polar of X₃ with respect to H_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 X₂₇₈. (Randy Hutson, September 14, 2016)