X₂₀₃₉

1st NINE-POINT-CIRCLE-KIEPERT-ASYMPTOTES INTERSECTION

Of the points other than X₁₁₅ in which the nine-point circle meets the asymptotes of the Kiepert hyperbola, X₂₀₃₉ is the one nearer to X₃.

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

X₂₀₃₉ (1st 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: {4, 1380}

• Related Centers: X₄ (ORTHOCENTER), X₁₃₈₀ (2nd BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE)

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): \(4.085713839267\)

▸ Linear Combinations1 combination

• X₂₀₃₉ = \(3\) X₃₈₁ + X₃₈₅₉₆, \(5\) X₁₆₅₆ - X₃₈₅₉₇, X₁₄₅₀₂ - \(3\) X₃₆₅₁₉

X₂₀₃₉ lies on the nine-point circle.

▸ Incident Central Lines33 lines

{2, 1379}, {3, 1349}, {4, 1380}, {5, 141}, {6, 1348}, {10, 2037}, {114, 3414}, {115, 2028}, {187, 19660}, {381, 38597}, {485, 1667}, {486, 1666}, {1341, 7790}, {1506, 2029}, {1656, 38596}, {1662, 1677}, {1663, 1676}, {1670, 2567}, {1671, 2566}, {2009, 2027}, {2010, 2026}, {2038, 2051}, {2542, 7797}, {2574, 13870}, {3557, 7752}, {5025, 13326}, {5996, 13636}, {6039, 31862}, {6189, 14568}, {13325, 37446}, {14501, 36519}, {18860, 19659}, {22242, 35913}

▸ Functional Relations13 relations

• X₂₀₃₉ = midpoint of X₄ and X₁₃₈₀

• X₂₀₃₉ = reflection of X_i in X_j for these {i,j}: {2029, 14633}, {2040, 5}

• X₂₀₃₉ = complement of X₁₃₇₉

• X₂₀₃₉ = complement of the isogonal conjugate of X₃₄₁₃

• X₂₀₃₉ = medial-isogonal conjugate of X₃₄₁₃

• X₂₀₃₉ = orthic-isogonal conjugate of X₃₄₁₃

• X₂₀₃₉ = X₄-Ceva conjugate of X₃₄₁₃

• X₂₀₃₉ = crosspoint of X₃₄₁₃ and X₁₄₆₃₃

• X₂₀₃₉ = crosssum of X₁₃₇₉ and X₁₄₆₃₁

• X₂₀₃₉ = crossdifference of every pair of points on line {3050, 41881}

• X₂₀₃₉ = X_i-complementary conjugate of X_j for these (i,j): {1, 3413}, {661, 39022}, {1380, 14838}, {1577, 2040}, {3413, 10}, {5638, 37}, {6189, 4369}, {13636, 8287}

• X₂₀₃₉ = barycentric quotient X_i/X_j for these {i,j}: {34981, 2040}, {39023, 6177}

• X₂₀₃₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {141, 625, 2040}, {623, 624, 2040}, {626, 5031, 2040}, {639, 32432, 2040}, {640, 32435, 2040}, {3454, 20546, 2040}, {3934, 5103, 2040}, {7684, 7685, 2040}, {11675, 39506, 2040}

Notes

The points X₂₀₃₉ and X₂₀₄₀ are endpoints of the diameter of the nine-point circle that is parallel to the line X₂₅X₃₉₄. (M. Iliev, 5/13/07)