X₂₂₂₂

X₂-ISOCONJUGATE OF X₆₅₄

X₂₂₂₂ = inverse-in-Stevanovic-circle of X₁₀₁

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

X₂₂₂₂ (X₂-ISOCONJUGATE OF X₆₅₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {484, 3465}

• Related Centers: X₄₈₄ (1st EVANS PERSPECTOR), X₃₄₆₅ (-CEVA CONJUGATE OF X₄₈₄)

Barycentrics: \(a(a - b)(a - c)(a - b + c)(a + b - c)(a^{2} - b^{2} + c^{2} - ac)(a^{2} - c^{2} + b^{2} - ab) : :\)

Search Value (6, 9, 13): \(-0.342354250614\)

X₂₂₂₂ lies on the circumcircle, the cubic K230.

▸ Incident Central Lines52 lines

{1, 953}, {3, 2716}, {12, 3109}, {35, 2687}, {36, 80}, {40, 2745}, {55, 2717}, {56, 2718}, {57, 840}, {59, 110}, {74, 484}, {98, 5143}, {99, 4998}, {100, 522}, {101, 650}, {102, 517}, {103, 1155}, {105, 2006}, {106, 1168}, {108, 7649}, {109, 513}, {112, 7115}, {162, 933}, {171, 2699}, {200, 2750}, {243, 917}, {651, 4588}, {661, 2149}, {675, 1447}, {741, 5061}, {759, 859}, {910, 2161}, {915, 1785}, {919, 1024}, {934, 3676}, {972, 5537}, {1141, 2166}, {1295, 2077}, {1298, 1956}, {1308, 2283}, {1309, 4242}, {1376, 2751}, {1477, 3660}, {1793, 5080}, {2249, 2341}, {2734, 6796}, {2739, 7580}, {2747, 5285}, {2756, 5687}, {4552, 9070}, {5089, 9085}, {5193, 8686}

▸ Functional Relations15 relations

• X₂₂₂₂ = midpoint of X₄₈₄ and X₃₄₆₅

• X₂₂₂₂ = reflection of X₂₇₁₆ in X₃

• X₂₂₂₂ = isogonal conjugate of X₃₇₃₈

• X₂₂₂₂ = cevapoint of X_i and X_j for these (i,j): (6, 8648), (513, 1319), (661, 3724), (663, 2183)

• X₂₂₂₂ = X_i-cross conjugate of X_j for these (i,j): (513, 1168), (1635, 57), (1769, 1), (2173, 7128), (3724, 2149), (5172, 59), (8648, 6)

• X₂₂₂₂ = inverse-in-Stevanovic-circle of X₁₀₁

• X₂₂₂₂ = trilinear pole of the line X₆X₁₄₁₁

• X₂₂₂₂ = {X(10016,X₁₀₀₁₇}-harmonic conjugate of X₂₇₁₆

• X₂₂₂₂ = concurrence of reflections in sides of ABC of line X₄X₈₀

• X₂₂₂₂ = Ψ(X₁, X₅)

• X₂₂₂₂ = Ψ(X₄, X₈₀)

• X₂₂₂₂ = Λ(X₈₀, X₃₇₆₂)

• X₂₂₂₂ = reflection of X₁₀₉ in line X₁X₃

• X₂₂₂₂ = trilinear product of circumcircle intercepts of line X₁X₅

• X₂₂₂₂ = X_i-isoconjugate of X_j for these (i,j): {1, 3738}, {2, 654}, {6, 3904}, {9, 3960}, {36, 522}, {54, 6369}, {55, 4453}, {60, 6370}, {75, 8648}, {284, 4707}, {320, 663}, {513, 4511}, {514, 2323}, {521, 1870}, {526, 3615}, {650, 3218}, {693, 2361}, {758, 3737}, {1443, 3900}, {1459, 5081}, {1464, 7253}, {1577, 4282}, {1983, 4858}, {2167, 2600}, {2170, 4585}, {2185, 2610}, {2245, 4560}, {3936, 7252}, {3939, 4089}, {4242, 7004}, {4391, 7113}

Notes

Let J_aJ_bJ_c be the excentral triangle in the plane of a triangle ABC. Let

• A″ = X₁₁₀-of-J_aBC, and define B″ and C″ cyclically

• (O_a) = circle with diameter AA″, and define (O_b) and (O_c) cyclically

The circles (O_a), (O_b), (O_c), and the circumcircle concur in X₂₂₂₂. See Hyacinthos 24369. (Antreas Hatzipolakis and Angel Montesdeoca, July 12, 2016)