X₂₉₆

ISOGONAL CONJUGATE OF X₂₄₃

X₂₉₆ is the isogonal conjugate of X₂₄₃.

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

X₂₉₆ (ISOGONAL CONJUGATE OF X₂₄₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {29, 65}, {1, 185}

• Related Centers: X₂₄₃ (-LINE CONJUGATE OF X₇₃), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER), X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE), X₁ (INCENTER), X₁₈₅ (NAGEL POINT OF THE ORTHIC TRIANGLE)

Trilinears: \((\cos A)/[\cos^{2} A - \cos B \cos C] : :\)

Barycentrics: \(a^{2} (a + b - c) \cdot (a - b + c) \cdot (a^{2} - b^{2} - c^{2}) \cdot (a^{2} b^{2} - b^{4} + a^{3} c - a b^{2} c - 2 a^{2} c^{2} + b^{2} c^{2} + a c^{3}) \cdot (a^{3} b - 2 a^{2} b^{2} + a b^{3} + a^{2} c^{2} - a b c^{2} + b^{2} c^{2} - c^{4}) : :\)

Search Value (6, 9, 13): \(1.490824291403\)

▸ Alternate Coordinates1 form

• Barycentrics: \((\sin 2A)/(\cos^{2} A - \cos B \cos C) : :\)

X₂₉₆ lies on the conic {A, B, C,X₁,X₃}}, the cubics K040 and K1185.

▸ Incident Central Lines26 lines

{1, 185}, {3, 820}, {4, 41500}, {29, 65}, {46, 2636}, {77, 1364}, {78, 7066}, {109, 284}, {219, 1949}, {243, 8764}, {283, 1813}, {416, 2651}, {518, 14198}, {851, 2655}, {916, 1807}, {949, 19350}, {1069, 20764}, {1155, 23707}, {1935, 7016}, {2263, 2818}, {3466, 5018}, {8677, 23696}, {10570, 15556}, {17973, 17975}, {38248, 38284}, {40442, 40946}

▸ Functional Relations12 relations

• X₂₉₆ = isogonal conjugate of X₂₄₃

• X₂₉₆ = isotomic conjugate of the polar conjugate of X₁₉₄₅

• X₂₉₆ = isogonal conjugate of the polar conjugate of X₁₉₅₂

• X₂₉₆ = X_i-Ceva conjugate of X_j for these (i,j): {1952, 1945}, {37142, 1937}

• X₂₉₆ = X_i-cross conjugate of X_j for these (i,j): {1936, 7016}, {8763, 1}

• X₂₉₆ = X_i-isoconjugate of X_j for these (i,j): {1, 243}, {2, 2202}, {4, 1936}, {6, 1948}, {8, 1430}, {19, 1944}, {29, 851}, {33, 5088}, {34, 7360}, {65, 15146}, {92, 1951}, {296, 41500}, {318, 26884}, {393, 6518}, {450, 7105}, {522, 23353}, {650, 1981}, {1172, 8680}, {2299, 44150}, {2322, 51647}, {2648, 41499}, {7016, 41497}, {31623, 42669}, {44112, 44130}

• X₂₉₆ = X_i-Dao conjugate of X_j for these (i,j): {3, 243}, {6, 1944}, {9, 1948}, {226, 44150}, {1936, 36033}, {1951, 22391}, {2202, 32664}, {7360, 11517}, {15146, 40602}

• X₂₉₆ = cevapoint of X_i and X_j for these (i,j): {1, 2655}, {3, 17975}, {1935, 1936}

• X₂₉₆ = crosssum of X₄₁₅ and X₂₆₅₉

• X₂₉₆ = trilinear pole of line {73, 652}

• X₂₉₆ = barycentric product X_i*X_j for these {i,j}: {1, 40843}, {3, 1952}, {63, 1937}, {69, 1945}, {73, 35145}, {75, 1949}, {307, 2249}, {520, 41207}, {656, 41206}, {1214, 37142}, {1942, 1943}

• X₂₉₆ = barycentric quotient X_i/X_j for these {i,j}: {1, 1948}, {3, 1944}, {6, 243}, {31, 2202}, {48, 1936}, {73, 8680}, {109, 1981}, {184, 1951}, {219, 7360}, {222, 5088}, {255, 6518}, {284, 15146}, {604, 1430}, {1214, 44150}, {1409, 851}, {1410, 51647}, {1415, 23353}, {1937, 92}, {1942, 7108}, {1945, 4}, {1949, 1}, {1950, 450}, {1952, 264}, {2202, 41500}, {2249, 29}, {2655, 39036}, {6516, 15418}, {7120, 41497}, {17966, 41499}, {17975, 39035}, {35145, 44130}, {37142, 31623}, {40843, 75}, {41206, 811}, {41207, 6528}