X₃₂₁₆

X₅₈-CEVA CONJUGATE OF X₁

X₃₂₁₆ = midpoint of X₁₂₀₁ and X₃₂₁₄

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

X₃₂₁₆ (X₅₈-CEVA CONJUGATE OF X₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1201, 3214}

• Related Centers: X₁₂₀₁ (3rd SARAGOSSA POINT OF X₅₆), X₃₂₁₄ (-CEVA CONJUGATE OF X₃₇)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₅₈ and \(u : v : w\) = X₁

Search Value (6, 9, 13): \(-0.231616617914\)

▸ Alternate Coordinates1 form

• Trilinears: \(a^{2}(b + c) + a(b^{2} + c^{2}) - bc(b + c) (E. Danneels)\)

X₃₂₁₆ lies on on the Feuerbach of the tangential triangle.

▸ Incident Central Lines117 lines

{1, 2}, {3, 1724}, {4, 5400}, {6, 474}, {9, 4261}, {21, 4256}, {35, 238}, {36, 5247}, {38, 3678}, {39, 2238}, {44, 3916}, {46, 2390}, {56, 4551}, {58, 404}, {65, 1739}, {72, 3670}, {73, 3911}, {100, 595}, {140, 5396}, {155, 6911}, {171, 1203}, {213, 1575}, {216, 1713}, {244, 3874}, {267, 2640}, {284, 7523}, {392, 4646}, {405, 4255}, {411, 1780}, {475, 3192}, {500, 549}, {517, 3987}, {518, 3953}, {579, 610}, {580, 6905}, {581, 631}, {602, 6796}, {662, 849}, {748, 5248}, {872, 6533}, {960, 4424}, {966, 5105}, {970, 1764}, {979, 2163}, {982, 5904}, {986, 5692}, {991, 3523}, {992, 2092}, {1015, 3780}, {1018, 2176}, {1045, 5506}, {1046, 1054}, {1050, 5541}, {1064, 6684}, {1104, 5440}, {1191, 5687}, {1213, 5153}, {1376, 5264}, {1450, 10106}, {1453, 5438}, {1457, 4848}, {1491, 4040}, {1498, 1754}, {1574, 2295}, {1654, 13571}, {1730, 10974}, {1738, 12047}, {1740, 2228}, {1757, 4283}, {1788, 10571}, {1834, 4187}, {1964, 4974}, {2077, 3073}, {2108, 6196}, {2234, 4672}, {2276, 3294}, {2292, 10176}, {2594, 5433}, {2650, 5883}, {2901, 4358}, {2915, 2916}, {2940, 2959}, {3191, 3772}, {3264, 3875}, {3290, 3970}, {3338, 3751}, {3454, 4202}, {3555, 4694}, {3570, 7760}, {3666, 5044}, {3684, 5299}, {3725, 4647}, {3740, 4719}, {3876, 4850}, {3878, 4642}, {3915, 8715}, {4003, 4005}, {4188, 4257}, {4205, 5241}, {4210, 4278}, {4300, 10164}, {4306, 5435}, {4413, 5711}, {4507, 9433}, {4653, 5047}, {4674, 5903}, {5255, 5315}, {5398, 6924}, {5706, 6918}, {5710, 9709}, {5713, 6854}, {5718, 8728}, {5721, 6922}, {9363, 13370}, {9548, 10882}, {9549, 12435}, {9567, 10441}

▸ Functional Relations14 relations

• X₃₂₁₆ = midpoint of X₁₂₀₁ and X₃₂₁₄

• X₃₂₁₆ = reflection of X₁ in X₁₂₀₁

• X₃₂₁₆ = X_i-Ceva conjugate of X_j for these (i,j): {58, 1}, {404, 3}

• X₃₂₁₆ = X₃₁₅₉-cross conjugate of X₁

• X₃₂₁₆ = crosspoint of X₆₆₂ and X₇₀₃₅

• X₃₂₁₆ = crossdifference of every pair of points on line (649, 3726)

• X₃₂₁₆ = crosssum of X_i and X_j for these (i,j): {513, 8054}, {661, 3248}

• X₃₂₁₆ = X_i-aleph conjugate of X_j for these (i,j): {1, 191}, {6, 1045}, {21, 20}, {28, 1714}, {58, 3216}, {81, 2}, {100, 4427}, {174, 1762}, {259, 2938}, {266, 1046}, {365, 846}, {366, 1761}, {509, 1781}, {662, 3882}, {6727, 3}

• X₃₂₁₆ = X₃₆₉₉-beth conjugate of X₃₂₁₆

• X₃₂₁₆ = X₇₄₁-he conjugate of X₆

• X₃₂₁₆ = X_i-zayin conjugate of X_j for these (i,j): {56, 1724}, {58, 3216}, {667, 4040}, {1193, 1}, {1203, 3293}, {2260, 1743}, {2308, 43}, {2309, 87}, {3122, 9359}, {3733, 3737}, {4057, 513}, {10457, 58}

• X₃₂₁₆ = barycentric product X₈₁*X₃₁₅₉

• X₃₂₁₆ = barycentric quotient X₃₁₅₉/X₃₂₁

• X₃₂₁₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (1, 43, 3293), (1, 6048, 3679), (2, 386, 1), (2, 9534, 10479), (3, 4383, 1724), (8, 995, 1), (10, 1193, 1), (42, 1125, 1), (43, 978, 1), (72, 3752, 3670), (614, 3811, 1), (899, 1193, 10), (936, 2999, 1), (975, 5256, 1), (1046, 1054, 3336), (1149, 3244, 1), (1698, 5313, 1), (3624, 5312, 1), (3751, 11512, 3338), (4202, 5741, 3454), (9534, 10479, 3679)

Notes

Let M_aM_bM_c be the medial triangle and A″B″C″ then cevian triangle of X₁. Let U be the circumcircle of A, M_a, A″, and define V and W cyclically. Then X₃₂₁₆ is the radical center of U, V, W. See Antreas Hatzipolakis and Peter Moses, Hyacinthos 267292.