X₃₅₇₁

INTERSECTION OF LINES P(6)P(8) AND U(6)U(8)

The points given by trilinears P(6) =\(b : c : a\) and U(6) =\(c : a : b\) are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) =\(1/b : 1/c : 1/a\) and U(8) =\(1/c : 1/a : 1/b\). The bicentric pair of lines P(6)P(8) and U(6)U(8) concur in X₃₅₇₁. Contributed by Peter Moses, July 7, 2009.

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

X₃₅₇₁ (INTERSECTION OF LINES P(6)P(8) AND U(6)U(8)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {9, 43}, {1, 512}

• Related Centers: X₉ (MITTENPUNKT), X₄₃ (-CEVA CONJUGATE OF X₁), X₁ (INCENTER), X₅₁₂ (ISOGONAL CONJUGATE OF X₉₉)

Trilinears: \(a^{4}bc + a^{2}(b^{4} - b^{3}c - 2b^{2}c^{2} - bc^{3} + c^{4}) + b^{3}c^{3} : :\)

Search Value (6, 9, 13): \(1.617052739601\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines3 lines

{1, 512}, {9, 43}, {1621, 1964}

▸ Functional Relations2 relations

• X₃₅₇₁ = crossdifference of PU(90)

• X₃₅₇₁ = crossdifference of every pair of points on line X₂₂₃₈X₄₃₆₇

Notes

Continuing, the line P(6)P(8) is the tangent to the 1st bicentric of the Kiepert hyperbola at P(8), and line U(6)U(8) is the tangent to the 2nd bicentric of the Kiepert hyperbola at U(8). (Randy Hutson, March 25, 2016)