X₁₉₀

YFF PARABOLIC POINT

In unpublished notes, Yff has studied the parabola tangent to sidelines BC, CA, AB and having focus X₁₀₁. If A',B',C' are the respective points of tangency, then the lines AA', BB', CC' concur in X₁₉₀.

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

X₁₉₀ (YFF PARABOLIC POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 45}, {9, 75}

• Related Centers: X₆₄₉ (CROSSDIFFERENCE OF X₁ and X₂), X₂ (CENTROID), X₄₅ (-BETH CONJUGATE OF X₁), X₉ (MITTENPUNKT), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER)

Trilinears: \(bc/(b - c) : ca/(c - a) : ab/(a - b)\)

Barycentrics: \(1/(b - c) : 1/(c - a) : 1/(a - b)\)

Search Value (6, 9, 13): \(4.477033349373\)

X₁₉₀ lies on the Steiner circumellipse.

▸ Incident Central Lines35 lines

{1, 537}, {2, 45}, {6, 192}, {7, 344}, {8, 528}, {9, 75}, {10, 671}, {37, 86}, {40, 341}, {44, 239}, {63, 312}, {69, 144}, {71, 290}, {72, 1043}, {99, 101}, {100, 659}, {110, 835}, {162, 643}, {191, 1089}, {238, 726}, {320, 527}, {321, 333}, {329, 345}, {350, 672}, {513, 660}, {514, 1016}, {522, 666}, {644, 651}, {646, 668}, {649, 889}, {658, 1020}, {670, 799}, {789, 813}, {872, 1045}, {1222, 3057}

▸ Functional Relations24 relations

• X₁₉₀ = reflection of X_i in X_j for these (i,j): (239, 44), (335, 37), (673, 9), (903, 2)

• X₁₉₀ = isogonal conjugate of X₆₄₉

• X₁₉₀ = isotomic conjugate of X₅₁₄

• X₁₉₀ = complement of X₄₄₄₀

• X₁₉₀ = anticomplement of X₁₀₈₆

• X₁₉₀ = X_i-Ceva conjugate of X_j for these (i,j): (99, 100), (666, 3570)

• X₁₉₀ = cevapoint of X_i and X_j for these (i,j): (2, 514), (9, 522), (37, 513), (440, 525)

• X₁₉₀ = X_i-cross conjugate of X_j for these (i,j): (513, 86), (514, 2), (522, 75)

• X₁₉₀ = crosssum of X₅₁₂ and X₇₉₈

• X₁₉₀ = crossdifference of every pair of points on line X₁₀₁₅X₁₉₆₀

• X₁₉₀ = X_i-aleph conjugate of X_j for these (i,j): (2, 1052), (190, 1), (645, 411), (668, 63), (1016, 100)

• X₁₉₀ = X_i-beth conjugate of X_j for these (i,j): (9, 292), (190, 651), (333, 88), (645, 190), (646, 646), (1016, 190)

• X₁₉₀ = trilinear pole of the line X₁X₂

• X₁₉₀ = Steiner-circumellipse-antipode of X₉₀₃

• X₁₉₀ = barycentric product of PU(24)

• X₁₉₀ = crossdifference of PU(25)

• X₁₉₀ = trilinear product of PU(58)

• X₁₉₀ = Steiner-circumellipse-X₁-antipode of X₃₂₂₇

• X₁₉₀ = Steiner-circumellipse-X₆-antipode of X₃₂₂₆

• X₁₉₀ = perspector of ABC and tangential triangle (wrt excentral triangle) of hyperbola passing through X₁, X₉ and the excenters (the Jerabek hyperbola of the excentral triangle)

• X₁₉₀ = X₆-isoconjugate of X₅₁₃

• X₁₉₀ = perspector of ABC and vertex-triangle of Gemini triangles 5 and 6

• X₁₉₀ = ABC-to-Gemini-triangle-19 parallelogic center

• X₁₉₀ = areal center of cevian triangles of PU(24)

Notes

The line X₁₀₀X₁₉₀ is tangent to the Steiner circumellipse at X₁₉₀ and to the circumcircle at X₁₀₀. (Peter Moses, July 7, 2009)

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 X₈ 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. The triangle A'B'C' is perspective to ABC, and the perspector is X₁₉₀. (Randy Hutson, December 26, 2015)

Let I_a be the reflection of X₁ in the perpendicular bisector of BC, and define I_b and I_c cyclically. X₁₉₀ = X₂₃₈ of I_aI_bI_c. (Randy Hutson, December 26, 2015)

Let A₅B₅C₅ and A₆B₆C₆ be Gemini triangles 5 and 6, resp. Let L_A be the tangent at A to conic {A,B₅,C₅,B₆,C₆}}, and define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₉₀. (Randy Hutson, January 15, 2019)