X₁₅₆

X₅-OF-TANGENTIAL-TRIANGLE

X₁₅₆ is the {X₁₅₄,X₁₅₅}-harmonic conjugate of X₂₆. For a list of harmonic conjugates, click Tables at the top of this page.

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

X₁₅₆ (X₅-OF-TANGENTIAL-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {26, 155}

• Related Centers: X₂₆ (CIRCUMCENTER OF THE TANGENTIAL TRIANGLE), X₁₅₅ (EIGENCENTER OF ORTHIC TRIANGLE)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a[b^{2}y/v + c^{2}z/w - a^{2}x/u]\), \(u = u(A,B,C) = \sin 2A\), \(v = u(B,C,A)\), \(w = u(C,A,B)\); \(x = x(A,B,C) = u^{2}(v^{2} + w^{2}) - (v^{2} - w^{2})^{2}\), \(y = x(B,C,A)\), \(z = x(C,A,B)\)

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

Search Value (6, 9, 13): \(-1.190548227388\)

▸ Incident Central Lines8 lines

{3, 74}, {4, 49}, {5, 184}, {25, 143}, {26, 154}, {54, 381}, {546, 578}, {550, 1092}

▸ Functional Relations4 relations

• X₁₅₆ = midpoint of X₂₆ and X₁₅₅

• X₁₅₆ = X₅-of-tangential-triangle

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

• X₁₅₆ = intouch-to-ABC functional image of X₅

Notes

Let O_A be the reflection of X₃ in BC, and define O_B and O_C cyclically. Let O'_A be the circumcenter of AO_BO_C, and define O'_B and O'_C cyclically. X₁₅₆ is the circumcenter of O'_AO'_BO'_C. (Randy Hutson, July 11, 2019)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). Then X₁₅₆ = X₅-of-A'B'C'. (Randy Hutson, July 11, 2019)