X₁₆₁₄

TCC-PERSPECTOR OF X₅₄

> X₁₆₁₄ lies on these lines: 3,74 4,184 6,1173 20,1147 22,155 23,52 24,154 30,49 51,1199 70,1176 185,186 376,1092 378,1498 389,1495 546,567 1503,1594

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

X₁₆₁₄ (TCC-PERSPECTOR OF X₅₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Trilinears: \(a[b^{2} \cos^{2}(C - A) + c^{2} \cos^{2}(A - B) - a^{2} \cos^{2}(B - C)] : :\)

Barycentrics: \(a^{2} (a^{8} - 3 a^{6} (b^{2} + c^{2}) + a^{4} (3 b^{4} + b^{2} c^{2} + 3 c^{4}) - a^{2} (b^{2} - c^{2})^{2} (b^{2} + c^{2}) + b^{2} c^{2} (b^{2} - c^{2})^{2}) : :\)

Search Value (6, 9, 13): \(-5.619839434332\)

▸ Functional Relations7 relations

• X₁₆₁₄ = isogonal conjugate of X₆₆₆₂

• X₁₆₁₄ = Vu tangential transform of X₃

• X₁₆₁₄ = X₁₂-of-tangential-triangle if ABC is acute

• X₁₆₁₄ = tangential isogonal conjugate of X₁₆₀₁

• X₁₆₁₄ = {X₃,X₁₅₆}-harmonic conjugate of X₁₁₀

• X₁₆₁₄ = exsimilicenter of circumcircle and nine-point circle of tangential triangle

• X₁₆₁₄ = intouch-to-ABC functional image of X₁₂

Notes

Let P₁ and P₂ be the two points on the circumcircle whose Steiner lines are tangent to the circumcircle. X₁₆₁₄ is the crosspoint of P₁ and P₂. (Randy Hutson, August 29, 2018)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anti-Euler triangle at X₁₆₁₄. Also, X₁₆₁₄ is the exsimilicenter of the circumcircle of ABC and nine-point circle of A'B'C'. (Randy Hutson, August 29, 2018)