X₂₀₇₀

INVERSE-IN-CIRCUMCIRCLE OF X₅

X₂₀₇₀ = midpoint of X₂₃ and X₁₈₆

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

X₂₀₇₀ (INVERSE-IN-CIRCUMCIRCLE OF X₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {23, 186}, {2, 3} (Euler line)

• Related Centers: X₂₃ (FAR-OUT POINT), X₁₈₆ (INVERSE-IN-CIRCUMCIRCLE OF X₄), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \((J^{2} - 2) \cos A + 4 \cos B \cos C\), where J is as at X₁₁₁₃

Search Value (6, 9, 13): \(0.225076619482\)

▸ Linear Combinations1 combination

• X₂₀₇₀ = X₃ + \(2\) X₂₃

▸ Incident Central Lines13 lines

{2, 3}, {49, 52}, {51, 567}, {54, 143}, {98, 1287}, {110, 1154}, {184, 568}, {187, 2079}, {231, 1989}, {399, 1495}, {476, 1141}, {500, 501}, {827, 842}

▸ Functional Relations16 relations

• X₂₀₇₀ = midpoint of X₂₃ and X₁₈₆

• X₂₀₇₀ = reflection of X_i in X_j for these (i,j): (3, 186), (2072, 468), (18859, 3)

• X₂₀₇₀ = isogonal conjugate of X₃₃₅₆₅

• X₂₀₇₀ = anticomplement of X₃₇₉₃₈

• X₂₀₇₀ = circumcircle-inverse of X₅

• X₂₀₇₀ = X₉₄-Ceva conjugate of X₆

• X₂₀₇₀ = crosspoint of X_i and X_j for these (i,j): (250, 476), (1141, 1166)

• X₂₀₇₀ = crosssum of X_i and X_j for these (i,j): (125, 526), (1154, 1209)

• X₂₀₇₀ = center of circle {{X₂₃,X₁₈₆,PU(4)}}, which is the inverse-in-circumcircle-of-orthocentroidal-circle

• X₂₀₇₀ = inverse-in-tangential-circle of X₃

• X₂₀₇₀ = pole wrt circumcircle of line PU(5) (line X₅X₅₂₃)

• X₂₀₇₀ = tangential isogonal conjugate of X₃₉₉

• X₂₀₇₀ = X₃₆-of-tangential-triangle if ABC is acute

• X₂₀₇₀ = {X₃,X₂₆}-harmonic conjugate of X₂₉₃₇

• X₂₀₇₀ = {X₃,X₂₃}-harmonic conjugate of X₃₇₉₂₄

• X₂₀₇₀ = crossdifference of every pair of points on line X₅₇₀X₆₄₇

Notes

As a point on the Euler line, X₂₀₇₀ has Shinagawa coefficients \((J^{2} - 2, -J^{2} + 6)\).