X₃₆₅₁

KS(1st CIRCUMPERP TRIANGLE)

For a construction of X₃₆₅₁ as the orthocenter of a certain triangle, see: Antreas Hatzipolakis and César Lozada, Hyacinthos 29288.

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

X₃₆₅₁ (KS(1st CIRCUMPERP TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 2475}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₄₇₅ (ANTICOMPLEMENT OF X₂₁), X₃ (CIRCUMCENTER)

Trilinears: \(abcph + a^{2}h^{2} - p(a - b)(a - c)[2abc + (b + c)(c + a - b)(a + b - c)]\), where \(p = a + b + c\), \(h = b^{2} + c^{2} - a^{2}\)

Search Value (6, 9, 13): \(10.387558928078\)

▸ Linear Combinations4 combinations

• X₃₆₅₁ = \(3R\) X₂ - \((6R + 2r)\) X₃

• X₃₆₅₁ = \(3\) X₃ - \(2\) X₅₄₂₈

• X₃₆₅₁ = \(3\) X₂₁ - \(4\) X₅₄₂₈

• X₃₆₅₁ = \(3\) X₁₆₅ - X₁₉₁

▸ Incident Central Lines42 lines

{2, 3}, {4, 442}, {9, 2173}, {35, 79}, {36, 950}, {40, 758}, {55, 3487}, {56, 3488}, {72, 74}, {78, 3587}, {81, 500}, {98, 1292}, {104, 4297}, {108, 1294}, {165, 191}, {201, 3465}, {212, 1745}, {329, 3648}, {477, 1290}, {515, 5258}, {581, 1754}, {601, 1742}, {842, 2691}, {944, 3428}, {1030, 1901}, {1058, 1617}, {1260, 3650}, {1330, 1792}, {1444, 2893}, {1612, 3772}, {1768, 2949}, {1936, 4303}, {2635, 3074}, {2693, 2766}, {2975, 3419}, {3072, 4300}, {3652, 5777}, {4333, 5010}, {5204, 5427}, {5217, 5714}, {5584, 5657}, {5752, 5890}

▸ Functional Relations6 relations

• X₃₆₅₁ = midpoint of X₂ and X₂₄₇₅

• X₃₆₅₁ = reflection of X_i in X_j for these (i,j): (4, 442), (21, 3)

• X₃₆₅₁ = X₇₈-gimel conjugate of X₃₄₃₀

• X₃₆₅₁ = reflection of X₂₁ in X₃

• X₃₆₅₁ = X₁₉₅-of-2nd-extouch-triangle

• X₃₆₅₁ = X_i-zayin conjugate of X_j for these (i,j): (9, 71), (522, 6003), (3465, 6000)

Notes

As a point on the Euler line, X₃₆₅₁ has Shinagawa coefficients \((2\$a\$E + 2abc, - 3\$a\$E - 2abc)\).