X₁₄₃

NINE-POINT CENTER OF ORTHIC TRIANGLE

X₁₄₃ is the QA-P13 center (Nine-point Center of the QA-Diagonal Triangle) of quadrangle ABCX₄ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/41-qa-p13.html). Also, X₁₄₃ is the QA-P22 center (Midpoint QA-P₁ and QA-P20) of quadrangle ABCX₄ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/50-qa-p22.html)

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

X₁₄₃ (NINE-POINT CENTER OF ORTHIC TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Orthic Triangle △H_aH_bH_c — vertices H_a, H_b, H_c (Pedal triangle of Orthocenter X₄ (altitude feet))

• Concurrence Lines: H_a–X₁₄₃, H_b–X₁₄₃, H_c–X₁₄₃ concurring at X₁₄₃

Trilinears: \((\sec A)[\cos(2C - 2A) + \cos(2A - 2B)] : :\)

Barycentrics: \((\tan A)[\cos(2C - 2A) + \cos(2A - 2B)] : :\)

Search Value (6, 9, 13): \(-0.095497496756\)

▸ Alternate Coordinates3 forms

• Trilinears: \((1 - 2 \cos 2A)\cos(B - C)]: :\)

• Trilinears: \(\sec A \cos(3A) \cos(B - C) : :\)

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

▸ Linear Combinations1 combination

• X₁₄₃ = X₅ - \(3\) X₅₁ = \(3\) X₅₁ + X₅₂ = X₄ + \(3\) X₅₆₈ = X₃ + \(3\) X₃₀₆₀ = \(3\) X₂₉₇₉ - \(7\) X₃₅₂₆ = X₃ - \(5\) X₃₅₆₇ = \(3\) X₃₀₆₀ + \(5\) X₃₅₆₇ = \(5\) X₆₃₂ - \(3\) X₃₉₁₇ = \(3\) X₁₄₀ - \(2\) X₅₄₄₇ = X₅₄₄₇ - \(3\) X₅₄₆₂ = \(3\) X₅ - X₅₅₆₂ = \(9\) X₅₁ - X₅₅₆₂ = \(3\) X₅₂ + X₅₅₆₂ = \(5\) X₁₆₅₆ - \(9\) X₅₆₄₀ = \(3\) X₃₈₁ - X₅₈₇₆ = \(3\) X₃₈₁ + X₅₈₈₉ = X₃₈₂ + \(3\) X₅₈₉₀ = \(5\) X₅₅₆₂ - \(9\) X₅₈₉₁ = \(5\) X₅ - \(3\) X₅₈₉₁ = \(5\) X₅₁ - X₅₈₉₁ = \(5\) X₅₂ + \(3\) X₅₈₉₁ = \(2\) X₃₅₃₀ - \(3\) X₅₈₉₂ = X₁₂₁₆ - \(3\) X₅₉₄₃ = \(2\) X₃₆₂₈ - \(3\) X₅₉₄₃ = X₃ - \(3\) X₅₉₄₆ = \(5\) X₃₅₆₇ - \(3\) X₅₉₄₆ = \(3\) X₅₆₈ - X₆₁₀₂ = \(3\) X₃₈₃₀ + X₆₂₄₁ = \(3\) X₂ + X₆₂₄₃ = \(3\) X₅₀₉₃ + X₆₄₀₃ = X₁₉₅ + \(3\) X₇₇₃₀ = \(11\) X₅₀₇₀ - \(7\) X₇₉₉₉

X₁₄₃ lies on the curves K054, K416, K464, Q106.

▸ Incident Central Lines45 lines

{2, 6101}, {3, 1173}, {4, 94}, {5, 51}, {6, 26}, {23, 1199}, {25, 156}, {30, 389}, {49, 1493}, {54, 2070}, {61, 2912}, {62, 2913}, {110, 195}, {140, 511}, {182, 7525}, {185, 3627}, {324, 565}, {381, 5876}, {382, 5890}, {567, 7488}, {569, 7502}, {575, 7555}, {576, 1147}, {578, 1658}, {632, 3917}, {970, 7508}, {1181, 7530}, {1216, 3628}, {1351, 6642}, {1353, 1843}, {1656, 5640}, {1993, 7506}, {2392, 5885}, {2937, 5012}, {2979, 3526}, {3517, 5093}, {3530, 5892}, {3580, 5576}, {3830, 6241}, {3850, 5907}, {3853, 6000}, {5070, 7999}, {5609, 7545}, {6515, 7528}, {7517, 7592}

▸ Functional Relations11 relations

• X₁₄₃ = midpoint of X_i and X_j for these {i,j}: {4, 6102}, {5, 52}, {185, 3627}, {389, 5446}, {1353, 1843}, {1493, 6152}, {3060, 5946}, {5876, 5889}, {6101, 6243}

• X₁₄₃ = reflection of X_i in X_j for these {i,j}: {140, 5462}, {1216, 3628}, {5907, 3850}

• X₁₄₃ = isogonal conjugate of X₂₅₂

• X₁₄₃ = anticomplement of X₃₂₁₄₂

• X₁₄₃ = X₁₃₇-cross conjugate of X₁₅₁₀

• X₁₄₃ = X₅-of-orthic triangle

• X₁₄₃ = X₂₄₉-Ceva conjugate of X₁₆₂₅

• X₁₄₃ = X₁₃₇-cross conjugate of X₁₅₁₀

• X₁₄₃ = excentral-to-ABC functional image of X₅

• X₁₄₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 6243, 6101), (3, 3567, 5946), (4, 568, 6102), (49, 1994, 1493), (51, 52, 5), (54, 2070, 5944), (381, 5889, 5876), (1112, 6746, 4), (1216, 5943, 3628), (1994, 3518, 49), (3060, 3567, 3)

• X₁₄₃ = X_i-isoconjugate of X_j for these {i,j}: {1, 252}, {54, 2962}, {93, 2169}, {930, 2616}, {2167, 2963}, {2190, 3519}

Notes

X₁₄₃ is the third of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see X₃₅₆₇.

Let A'B'C' be the cevian triangle of X₅. Let A″, B″, C″ be the inverse-in-circumcircle of A', B', C'. The lines AA″, BB″, CC″ concur in X₁₄₃. Also, X₁₄₃ = intersection of the tangent to hyperbola {A, B, C,X₄,X₁₅}} at X₆₁ and the tangent to the hyperbola {A, B, C,X₄,X₁₆}} at X₆₂. (Randy Hutson, July 23, 2015)