X₃₃₉₈

COS(A - 2ω) POINT

X₃₃₉₈ and related points occupy Bernard Gibert’s Table 38. See the note at X₃₃₆₄.

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

X₃₃₉₈ (COS(A - 2ω) POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1342, 1343}

• Related Centers: X₁₃₄₂ (INSIMILICENTER), X₁₃₄₃ (EXSIMILICENTER)

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

Search Value (6, 9, 13): \(2.776456554581\)

▸ Alternate Coordinates1 form

• Trilinears: \(e^{2}\sin(A - \omega ) - \sin(A + \omega ) : :\)

▸ Incident Central Lines5 lines

{3, 6}, {5, 83}, {140, 325}, {384, 2782}, {597, 1576}

▸ Functional Relations9 relations

• X₃₃₉₈ = midpoint of X₁₃₄₂ and X₁₃₄₃

• X₃₃₉₈ = isogonal conjugate of X₃₃₉₉

• X₃₃₉₈ = inverse-in-Brocard-circle of X₃₀₉₅

• X₃₃₉₈ = crosssum of X₃₁₀₂ ande X₃₁₀₃

• X₃₃₉₈ = harmonic center of circumcircle and 1st Lemoine circle

• X₃₃₉₈ = harmonic center of 1st and 2nd Brocard circles

• X₃₃₉₈ = X₃-of-5th-anti-Brocard-triangle

• X₃₃₉₈ = radical center of circumcircles of ABC, 5th Brocard triangle, and 5th anti-Brocard triangle

• X₃₃₉₈ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (32, 182, 3), (371, 372, 3094), (1687, 1688, 39)

Notes

Randy Huston, September 5, 2015, gives three constructions of X₃₃₉₈:

1. Let X be the 2nd Brocard point of pedal triangle of 1st Brocard point. Let Y be the 1st Brocard point of pedal triangle of 2nd Brocard point. Then X₃₃₉₈ is the vertex conjugate of X and Y.

2. Let U be the circle {{X₃₇₁,X₃₇₂,PU(1),PU(39)}} and V the 2nd Brocard circle. Then X₃₃₉₈ is the center of the inverse-in-U of V.

3. Let A'B'C' be the 3rd Brocard triangle. Let O_a be the circumcenter of A'BC, and define O_b and O_c cyclically. The lines AO_a, BO_b, CO_c concur in X₃₃₉₈.