X₂₀₄₀

2nd NINE-POINT-CIRCLE-KIEPERT-ASYMPTOTES INTERSECTION

X₂₀₄₀ = complement of X₁₃₈₀

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

X₂₀₄₀ (2nd NINE-POINT-CIRCLE-KIEPERT-ASYMPTOTES INTERSECTION) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 141}, {3, 1348}

• Related Centers: X₅ (NINE-POINT CENTER), X₁₄₁ (COMPLEMENT OF SYMMEDIAN POINT), X₃ (CIRCUMCENTER), X₁₃₄₈ (INSIMILICENTER)

Trilinears: \(e \cos A + 2e \cos B \cos C - \cos(A + \omega )\)

Barycentrics: \((S_{B} + S_{C}) \cdot (S_{A}^{2} - S_{B} S_{C}) - (S^{2} + S_{B} S_{C}) \sqrt{-S^{2} + S_{A}^{2} + S_{B}^{2} + S_{C}^{2}} : :\)

Search Value (6, 9, 13): \(-2.979970355997\)

▸ Linear Combinations1 combination

• X₂₀₄₀ = \(3\) X₃₈₁ + X₃₈₅₉₆, \(5\) X₁₆₅₆ - X₃₈₅₉₇, X₁₄₅₀₂ - \(3\) X₃₆₅₁₉

X₂₀₄₀ lies on the nine-point circle.

▸ Incident Central Lines33 lines

{2, 1380}, {3, 1348}, {4, 1379}, {5, 141}, {6, 1349}, {10, 2038}, {114, 3413}, {115, 2029}, {187, 19659}, {381, 38596}, {485, 1666}, {486, 1667}, {1340, 7790}, {1506, 2028}, {1656, 38597}, {1662, 1676}, {1663, 1677}, {1670, 2566}, {1671, 2567}, {2009, 2026}, {2010, 2027}, {2037, 2051}, {2543, 7797}, {2575, 13870}, {3558, 7752}, {5025, 13325}, {5996, 13722}, {6040, 31863}, {6190, 14568}, {13326, 37446}, {14502, 36519}, {18860, 19660}, {22243, 35914}

▸ Functional Relations13 relations

• X₂₀₄₀ = midpoint of X_i and X_j for these {i,j}: {4, 1379}, {115, 14501}, {6040, 31863}, {6190, 31862}

• X₂₀₄₀ = reflection of X_i in X_j for these {i,j}: {2028, 14632}, {2039, 5}

• X₂₀₄₀ = complement of X₁₃₈₀

• X₂₀₄₀ = complement of the isogonal conjugate of X₃₄₁₄

• X₂₀₄₀ = medial-isogonal conjugate of X₃₄₁₄

• X₂₀₄₀ = orthic-isogonal conjugate of X₃₄₁₄

• X₂₀₄₀ = X₄-Ceva conjugate of X₃₄₁₄

• X₂₀₄₀ = crosspoint of X₃₄₁₄ and X₁₄₆₃₂

• X₂₀₄₀ = crosssum of X₁₃₈₀ and X₁₄₆₃₀

• X₂₀₄₀ = crossdifference of every pair of points on line {3050, 41880}

• X₂₀₄₀ = X_i-complementary conjugate of X_j for these (i,j): {1, 3414}, {661, 39023}, {1379, 14838}, {1577, 2039}, {3414, 10}, {5639, 37}, {6190, 4369}, {13722, 8287}

• X₂₀₄₀ = barycentric quotient X_i/X_j for these {i,j}: {34981, 2039}, {39022, 6178}

• X₂₀₄₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {141, 625, 2039}, {623, 624, 2039}, {626, 5031, 2039}, {639, 32432, 2039}, {640, 32435, 2039}, {3454, 20546, 2039}, {3934, 5103, 2039}, {7684, 7685, 2039}, {11675, 39506, 2039}