X₁₆₉₁

RADICAL TRACE OF CIRCUMCIRCLE AND 1st LEMOINE CIRCLE

X₁₆₉₁ is the perspector of ABC and the reflection of the tangential triangle in the Lemoine axis (i.e., the reflection of the anticevian triangle of X₆ in the trilinear polar of X₆). (Randy Hutson, September 5, 2015)

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

X₁₆₉₁ (RADICAL TRACE OF CIRCUMCIRCLE AND 1st LEMOINE CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Central Lines: {3, 6}, {98, 230}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₉₈ (TARRY POINT), X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄)

Trilinears: \(\sin(A - 2\omega ) : :\)

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

Search Value (6, 9, 13): \(-0.896286326706\)

▸ Alternate Coordinates1 form

• Trilinears: e^2 cos(A - ω) - cos(A + ω) : : (c.f. X₂₄₅₆)

▸ Incident Central Lines17 lines

{2, 1501}, {3, 6}, {31, 893}, {83, 316}, {98, 230}, {99, 698}, {141, 1078}, {154, 1611}, {184, 1613}, {237, 694}, {249, 524}, {385, 732}, {691, 729}, {695, 1176}, {699, 805}, {1428, 1914}, {1968, 1974}

▸ Functional Relations22 relations

• X₁₆₉₁ = midpoint of X_i and X_j for these (i,j): (6, 2076), (187, 1692), (1687, 1688)

• X₁₆₉₁ = reflection of X_i in X_j for these (i,j): (6, 1692), (1692, 2030), (2076, 187)

• X₁₆₉₁ = isogonal conjugate of X₁₉₁₆

• X₁₆₉₁ = isotomic conjugate of X₁₈₈₉₆

• X₁₆₉₁ = complement of X₅₂₀₇

• X₁₆₉₁ = inverse-in-circumcircle of X₃₂

• X₁₆₉₁ = inverse-in-Brocard circle of X₃₀₉₄

• X₁₆₉₁ = inverse-in-1st-Lemoine-circle of X₆

• X₁₆₉₁ = inverse-in-2nd-Lemoine-circle of X₅₇₆

• X₁₆₉₁ = X_i-Ceva conjugate of X_j for these (i,j): (237, 1971), (699, 32), (1976, 6)

• X₁₆₉₁ = crosspoint of X_i and X_j for these (i,j): (83, 98), (385, 419)

• X₁₆₉₁ = crosssum of X₃₉ and X₅₁₁

• X₁₆₉₁ = reflection of X₂₀₇₆ in the Lemoine axis

• X₁₆₉₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (32, 182, 6), (371, 372, 3095), (1342, 1343, 39)

• X₁₆₉₁ = crossdifference of every pair of points on the line PU(11)

• X₁₆₉₁ = center of the circle {{X₁₆₈₇,X₁₆₈₈,PU(1),PU(2)}} (the circle orthogonal to the circumcircle and passing through the 1st and 2nd Brocard points)

• X₁₆₉₁ = intersection of tangents to 1st Lemoine circle at intersections with 2nd Lemoine circle

• X₁₆₉₁ = centroid of X₆X₁₅X₁₆

• X₁₆₉₁ = X₁₆₉₁ of circumsymmedial triangle

• X₁₆₉₁ = harmonic center of 1st and 2nd Lemoine circles

• X₁₆₉₁ = harmonic center of 2nd Brocard circle and the circle {{X₃₇₁,X₃₇₂,PU(1),PU(39)}}

• X₁₆₉₁ = perspector of ABC and the reflection of the 2nd Ehrmann triangle in line X₆X₅₁₂ (the perspectrix of ABC and 2nd Ehrmann triangle)

Notes

Let A'B'C' be the 1st anti-Brocard triangle. X₁₆₉₁ is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, July 20, 2016)

Let A' be the circumcircle intercept, other than A, of the A-Montesdeoca-Lemoine circle. Define B' and C' cyclically. Triangle A'B'C' is perspective to the symmedial triangle at X₁₆₉₁. (Randy Hutson, July 11, 2019)