X₅₉₇

MIDPOINT OF X₂ AND X₆

X₅₉₇ is the midpoint of X₂ and X₆.

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

X₅₉₇ (MIDPOINT OF X₂ AND X₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}

• Related Centers: X₂ (CENTROID), X₆ (SYMMEDIAN POINT)

Trilinears: \((4a^{2} + b^{2} + c^{2})/a : (4b^{2} + c^{2} + a^{2})/b : (4c^{2} + a^{2} + b^{2})/c\)

Barycentrics: \(4a^{2} + b^{2} + c^{2} : 4b^{2} + c^{2} + a^{2} : 4c^{2} + a^{2} + b^{2}\)

Search Value (6, 9, 13): \(1.811138643777\)

▸ Incident Central Lines9 lines

{2, 6}, {5, 542}, {30, 182}, {39, 1084}, {83, 671}, {140, 576}, {373, 2854}, {511, 549}, {518, 551}

▸ Functional Relations16 relations

• X₅₉₇ = midpoint of X₂ and X₆

• X₅₉₇ = reflection of X₁₄₁ in X₂

• X₅₉₇ = isogonal conjugate of X₃₉₃₈₉

• X₅₉₇ = complement of X₅₉₉

• X₅₉₇ = crosssum of X₆ and X₅₇₄

• X₅₉₇ = circumcenter of the pedal triangle of X₂

• X₅₉₇ = circumcenter of the pedal triangle of X₆

• X₅₉₇ = crosspoint of X₂ and X₅₉₈

• X₅₉₇ = centroid of pedal triangle of X₁₈₂

• X₅₉₇ = crossdifference of every pair of points on line X₅₁₂X₅₁₀₄

• X₅₉₇ = centroid of set consisting of the interiors (with or without boundaries) of the power circles

• X₅₉₇ = {X₃₉₅,X₃₉₆}-harmonic conjugate of X₃₈₁₅

• X₅₉₇ = harmonic center of circles O(13,15) and O(14,16)

• X₅₉₇ = center of Lemoine ellipse

• X₅₉₇ = X₅-of-anti-Artzt-triangle

• X₅₉₇ = {X₂,X₁₉₉₂}-harmonic conjugate of X₅₉₉

Notes

Let O_A be the circle centered at the A-vertex of the antipedal triangle of X₂ and passing through A; define O_B and O_C cyclically. X₅₉₇ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)