X₂₇

CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER

X₂₇ is the cevapoint of orthocenter and Clawson center.

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

X₂₇ (CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {19, 63}

• Related Centers: X₄ (ORTHOCENTER), X₁₉ (CLAWSON POINT), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₆₃ (ISOGONAL CONJUGATE OF X₁₉)

Trilinears: \((\sec A)/(b + c) : (\sec B)/(c + a) : (\sec C)/(a + b)\)

Barycentrics: \((\tan A)/(b + c) : (\tan B)/(c + a) : (\tan C)/(a + b)\)

Search Value (6, 9, 13): \(-2.596826036189\)

▸ Linear Combinations1 combination

• X₂₇ = \(3 S_{A} S_{B} S_{C}\) X₂ - \(2 S^{2} s^{2}\) X₃

▸ Incident Central Lines175 lines

{2, 3}, {6, 1246}, {7, 81}, {8, 19848}, {19, 63}, {33, 5287}, {34, 5256}, {53, 37646}, {57, 273}, {58, 270}, {69, 19793}, {71, 40435}, {84, 1896}, {86, 1474}, {99, 9085}, {101, 22000}, {103, 107}, {110, 917}, {112, 675}, {116, 40076}, {162, 673}, {198, 27287}, {225, 2363}, {226, 284}, {239, 1829}, {240, 11031}, {242, 2355}, {243, 1859}, {264, 14829}, {272, 2189}, {274, 19798}, {275, 2051}, {281, 5235}, {295, 335}, {306, 1043}, {310, 17206}, {317, 4417}, {318, 10461}, {321, 5279}, {329, 2287}, {331, 15467}, {393, 967}, {394, 10446}, {516, 2328}, {579, 1751}, {607, 39741}, {648, 903}, {653, 18815}, {662, 913}, {811, 19801}, {823, 34234}, {871, 19800}, {908, 2327}, {933, 26708}, {946, 2360}, {1014, 1440}, {1019, 40213}, {1071, 1871}, {1086, 16099}, {1088, 1434}, {1119, 21454}, {1230, 44146}, {1240, 19810}, {1249, 37666}, {1259, 19845}, {1268, 1796}, {1301, 41905}, {1304, 2688}, {1333, 3772}, {1412, 17197}, {1427, 18603}, {1441, 3101}, {1625, 35096}, {1659, 2067}, {1699, 17188}, {1719, 1733}, {1730, 1746}, {1770, 1780}, {1785, 39595}, {1803, 2332}, {1810, 19815}, {1812, 17139}, {1836, 2194}, {1840, 40033}, {1841, 3666}, {1844, 10122}, {1851, 14024}, {1865, 18679}, {1870, 17011}, {1880, 25059}, {1882, 1940}, {1973, 2296}, {2052, 13478}, {2193, 37695}, {2203, 14621}, {2206, 3120}, {2221, 4000}, {2354, 28287}, {2400, 7192}, {2659, 26892}, {2969, 6650}, {3011, 19849}, {3187, 3868}, {3194, 5222}, {3306, 19802}, {3332, 11206}, {3423, 5324}, {3661, 5090}, {3687, 5081}, {3794, 17616}, {3914, 44119}, {4304, 4653}, {4373, 9965}, {4384, 5342}, {4393, 11396}, {4786, 7649}, {4921, 19819}, {5057, 6061}, {5088, 18607}, {5333, 17917}, {5523, 18686}, {5732, 17194}, {5905, 39695}, {6198, 17019}, {6331, 19816}, {6335, 40039}, {6358, 16548}, {6384, 19803}, {6502, 13390}, {6542, 12135}, {6548, 17925}, {6748, 37662}, {7017, 19807}, {7119, 40418}, {7140, 17927}, {7249, 40432}, {7283, 42706}, {7354, 40980}, {7718, 17316}, {8025, 19823}, {8044, 34440}, {8736, 37770}, {8756, 19797}, {9308, 37683}, {10444, 17185}, {10449, 19838}, {11363, 16826}, {13243, 35360}, {16077, 35161}, {16747, 20880}, {17182, 24556}, {17189, 23681}, {17277, 44103}, {17903, 41364}, {17921, 43925}, {18344, 24601}, {18742, 40010}, {19752, 19767}, {19799, 33932}, {19812, 25507}, {19820, 39710}, {19821, 29766}, {19824, 36606}, {19827, 28650}, {19830, 39707}, {20291, 38852}, {20527, 20751}, {21370, 44178}, {21621, 24019}, {23383, 34429}, {31424, 39585}, {32000, 37655}, {34255, 39749}, {39704, 42028}, {41342, 43729}

▸ Functional Relations20 relations

• X₂₇ = isogonal conjugate of X₇₁

• X₂₇ = isotomic conjugate of X₃₀₆

• X₂₇ = inverse-in-circumcircle of X₂₀₇₃

• X₂₇ = inverse-in-orthocentroidal-circle of X₄₆₉

• X₂₇ = complement of X₃₁₅₁

• X₂₇ = anticomplement of X₄₄₀

• X₂₇ = X₂₈₆-Ceva conjugate of X₂₉

• X₂₇ = cevapoint of X_i and X_j for these (i,j): (4, 19), (57, 278)

• X₂₇ = X_i-cross conjugate of X_j for these (i,j): (4, 286), (19, 28), (57, 81), (58, 86)

• X₂₇ = crossdifference of every pair of points on line X₆₄₇X₈₁₀

• X₂₇ = X_i-Hirst inverse of X_j for these (i,j): (2, 447), (4, 423)

• X₂₇ = X_i-beth conjugate of X_j for these (i,j): (648, 27), (923, 27)

• X₂₇ = trilinear pole of line X₂₄₂X₅₁₄ (the polar of X₁₀ wrt polar circle)

• X₂₇ = pole wrt polar circle of trilinear polar of X₁₀ (line X₅₂₃X₆₆₁)

• X₂₇ = polar conjugate of X₁₀

• X₂₇ = X₆-isoconjugate of X₇₂

• X₂₇ = X₁₉-isoconjugate of X₃₆₈₂

• X₂₇ = X₇₅-isoconjugate of X₂₂₀₀

• X₂₇ = crosspoint of X₄ and X₁₉ wrt excentral triangle

• X₂₇ = trilinear product X₂*X₂₈

Notes

As a point on the Euler line, X₂₇ has Shinagawa coefficients \((F, -E - F - \$bc\$)\).

X₂₇ is the {X₂,X₄}-harmonic conjugate of X₄₆₉.