X₆₅₄

CROSSDIFFERENCE OF X₁ AND X₅

X₆₅₄ = reflection of X₆₅₄ in the anti-orthic axis

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

X₆₅₄ (CROSSDIFFERENCE OF X₁ AND X₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {101, 109}, {44, 513}

• Related Centers: X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER)), X₄₄ (-LINE CONJUGATE OF X₁), X₅₁₃ (ISOGONAL CONJUGATE OF X₁₀₀)

Trilinears: \(\cos(A - B) - \cos(C - A) : :\)

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

Search Value (6, 9, 13): \(-12.173358731681\)

X₆₅₄ lies on the Terzic circle.

▸ Incident Central Lines52 lines

{3, 41155}, {6, 3310}, {9, 1639}, {44, 513}, {50, 647}, {55, 926}, {57, 1638}, {63, 918}, {81, 42744}, {101, 109}, {103, 2291}, {171, 24462}, {182, 41158}, {514, 23737}, {517, 41162}, {812, 40166}, {900, 2161}, {909, 2432}, {1021, 2596}, {1768, 2826}, {1946, 8676}, {2170, 4435}, {2352, 23225}, {2423, 14578}, {2594, 2598}, {2774, 41164}, {2786, 41163}, {2824, 41165}, {3196, 27780}, {3218, 4453}, {3219, 30565}, {3305, 45326}, {3306, 44902}, {3669, 4091}, {3887, 41166}, {4041, 42649}, {4063, 21120}, {4131, 28984}, {4282, 7252}, {4895, 42657}, {4984, 14400}, {5220, 30700}, {6589, 22383}, {8659, 41156}, {14418, 42552}, {17455, 21758}, {20295, 28834}, {22160, 44410}, {25924, 31250}, {32641, 32675}, {35365, 36057}, {36054, 43060}

▸ Functional Relations19 relations

• X₆₅₄ = reflection of X₅₅ in X₆₁₃₉

• X₆₅₄ = reflection of X₆₅₄ in the anti-orthic axis

• X₆₅₄ = isogonal conjugate of X₆₅₅

• X₆₅₄ = circumcircle-inverse of X₄₁₁₅₅

• X₆₅₄ = Brocard-circle-inverse of X₄₁₁₅₈

• X₆₅₄ = X₃₁-complementary conjugate of X₃₈₉₈₄

• X₆₅₄ = X_i-Ceva conjugate of X_j for these (i,j): {2, 38984}, {54, 41218}, {74, 3270}, {101, 17455}, {110, 215}, {651, 34586}, {655, 1}, {901, 55}, {1983, 2245}, {2250, 11998}, {3065, 2310}, {3461, 41211}, {3466, 2638}, {23838, 663}, {24029, 1319}, {24624, 11}, {32641, 6}, {40215, 3025}

• X₆₅₄ = X_i-cross conjugate of X_j for these (i,j): {2081, 2600}, {41218, 54}

• X₆₅₄ = cevapoint of X₂₀₈₁ and X₂₆₂₄

• X₆₅₄ = crosspoint of X_i and X_j for these (i,j): {1, 655}, {9, 5548}, {57, 2720}, {100, 2990}, {101, 2316}, {104, 651}, {110, 24624}, {275, 1309}, {901, 40215}, {1983, 4282}, {3738, 3960}, {3904, 44428}, {4560, 43728}

• X₆₅₄ = crosssum of X_i and X_j for these (i,j): {1, 654}, {6, 39200}, {9, 2804}, {57, 30725}, {216, 8677}, {513, 8609}, {514, 3911}, {517, 650}, {523, 2245}, {2599, 2600}, {4559, 23981}

• X₆₅₄ = crossdifference of every pair of points on line {1, 5}

• X₆₅₄ = perspector of hyperbola {A, B, C,X₁,X₃₆}}

• X₆₅₄ = bicentric difference of PU(68)

• X₆₅₄ = PU(68)-harmonic conjugate of X₂₅₉₄

• X₆₅₄ = X_i-isoconjugate of X_j for these (i,j): {1, 655}, {2, 2222}, {6, 35174}, {12, 37140}, {56, 36804}, {75, 32675}, {80, 651}, {100, 2006}, {101, 18815}, {109, 18359}, {190, 1411}, {476, 16577}, {653, 1807}, {664, 2161}, {759, 4552}, {901, 14628}, {934, 36910}, {1020, 6740}, {1415, 20566}, {2341, 4566}, {2594, 32680}, {3257, 14584}, {3738, 23592}, {4551, 24624}, {4554, 6187}, {4559, 14616}, {4565, 15065}, {4573, 34857}, {6358, 36069}, {14213, 36078}, {21741, 35139}, {24029, 40437}, {26700, 41226}, {32671, 34388}, {32678, 40999}, {34242, 44765}, {35307, 39277}, {37222, 37630}

• X₆₅₄ = barycentric product X_i*X_j for these {i,j}: {1, 3738}, {3, 44428}, {6, 3904}, {9, 3960}, {36, 522}, {54, 6369}, {55, 4453}, {60, 6370}, {75, 8648}, {214, 23838}, {261, 42666}, {284, 4707}, {312, 21758}, {318, 22379}, {320, 663}, {333, 21828}, {513, 4511}, {514, 2323}, {521, 1870}, {526, 3615}, {649, 32851}, {650, 3218}, {652, 17923}, {655, 35128}, {656, 17515}, {657, 17078}, {693, 2361}, {758, 3737}, {860, 23189}, {1021, 18593}, {1443, 3900}, {1459, 5081}, {1464, 7253}, {1577, 4282}, {1639, 40215}, {1845, 37628}, {1983, 4858}, {2167, 2600}, {2170, 4585}, {2185, 2610}, {2245, 4560}, {2364, 23884}, {3063, 20924}, {3064, 22128}, {3724, 18155}, {3936, 7252}, {3939, 4089}, {4242, 7004}, {4391, 7113}, {4768, 16944}, {21012, 39178}, {21789, 41804}, {34586, 43728}, {35011, 45950}

• X₆₅₄ = barycentric quotient X_i/X_j for these {i,j}: {1, 35174}, {6, 655}, {9, 36804}, {31, 2222}, {32, 32675}, {36, 664}, {320, 4572}, {513, 18815}, {522, 20566}, {526, 40999}, {649, 2006}, {650, 18359}, {657, 36910}, {663, 80}, {667, 1411}, {1443, 4569}, {1464, 4566}, {1635, 14628}, {1870, 18026}, {1946, 1807}, {1960, 14584}, {1983, 4564}, {2150, 37140}, {2245, 4552}, {2323, 190}, {2361, 100}, {2600, 14213}, {2610, 6358}, {2624, 16577}, {3025, 4453}, {3063, 2161}, {3218, 4554}, {3615, 35139}, {3724, 4551}, {3737, 14616}, {3738, 75}, {3904, 76}, {3960, 85}, {4041, 15065}, {4282, 662}, {4453, 6063}, {4511, 668}, {4707, 349}, {6369, 311}, {6370, 34388}, {7113, 651}, {7252, 24624}, {8648, 1}, {9404, 41226}, {14270, 2594}, {17515, 811}, {21758, 57}, {21789, 6740}, {21828, 226}, {22379, 77}, {32675, 23592}, {32851, 1978}, {34544, 4585}, {35128, 3904}, {41218, 6369}, {42666, 12}, {44428, 264}

• X₆₅₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {649, 652, 650}, {650, 652, 9404}, {650, 4790, 13401}, {2590, 2591, 650}, {2610, 2624, 2245}, {3310, 22086, 6}, {4394, 14298, 650}