X₂₀₄₈

{X₃,X₅}-HARMONIC CONJUGATE OF X₂₀₄₇

X₂₀₄₈ is the {X₃,X₅}-harmonic conjugate of X₂₀₄₇.

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

X₂₀₄₈ ({X₃,X₅}-HARMONIC CONJUGATE OF X₂₀₄₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 5} (Euler line)

• Related Centers: X₂₀₄₇ (EULER LINE INTERCEPT OF LINE X), X₃ (CIRCUMCENTER), X₅ (NINE-POINT CENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (s - r)\cos A - (s + r)\cos(B - C)\), where r = inradius, s = semiperimeter

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(-3.468071768266\)

▸ Incident Central Lines2 lines

{2, 3}, {486, 1686}

Notes

As a point on the Euler line, X₂₀₄₈ has Shinagawa coefficients \((r, s)\).