X₁₁₅₃

CENTER OF THE VAN LAMOEN CIRCLE

A triangle is divided by its three medians into 6 smaller triangles. The circumcenters of these smaller triangles are concyclic. Their circle, the Van Lamoen circle, is introduced in

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

X₁₁₅₃ (CENTER OF THE VAN LAMOEN CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 187}, {140, 524}

• Related Centers: X₂ (CENTROID), X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X), X₁₄₀ (MIDPOINT OF X₃ AND X₅), X₅₂₄ (ISOGONAL CONJUGATE OF X₁₁₁)

Trilinears: \(bc[13a^{2}(b^{2} + c^{2}) + 10b^{2}c^{2} - 10a^{4} - 4b^{4} - 4c^{4}] : :\)

Barycentrics: \(13a^{2}(b^{2} + c^{2}) + 10b^{2}c^{2} - 10a^{4} - 4b^{4} - 4c^{4} : :\)

Search Value (6, 9, 13): \(4.912556924389\)

▸ Alternate Coordinates1 form

• Barycentrics: \(10 a^{4} - 13 a^{2} b^{2} + 4 b^{4} - 13 a^{2} c^{2} - 10 b^{2} c^{2} + 4 c^{4} : :\)

▸ Linear Combinations2 combinations

• X₁₁₅₃ = X₂ + X₈₁₈₂, \(9\) X₂ - X₂₃₃₃₄, \(13\) X₂ - X₄₄₆₇₈, \(10\) X₁₄₀ - X₇₇₆₄, \(8\) X₁₄₀ + X₇₇₈₀, \(2\) X₁₄₀ + X₃₄₅₀₆, \(3\) X₅₄₉ + X₁₆₅₀₉, \(15\) X₆₃₁ + X₅₄₈₅, \(5\) X₆₃₁ - X₇₆₁₈, \(10\) X₆₃₂ - X₇₈₄₃, \(7\) X₃₅₂₃ + X₇₆₂₀, \(7\) X₃₅₂₃ - X₃₄₅₀₄, \(3\) X₃₅₂₄ + X₇₆₁₅, \(7\) X₃₅₂₆ - X₇₇₇₅, \(3\) X₅₀₅₄ + X₇₆₁₀, \(3\) X₅₀₅₄ - X₇₆₂₂, \(9\) X₅₀₅₄ - X₁₁₁₆₅, X₅₄₈₅ + \(3\) X₇₆₁₈, \(3\) X₅₅₆₉ + X₈₁₇₆, \(3\) X₅₅₆₉ - X₈₁₈₂, \(9\) X₅₅₆₉ + X₂₃₃₃₄, \(13\) X₅₅₆₉ + X₄₄₆₇₈, \(3\) X₇₆₀₆ - X₄₂₅₃₆, \(3\) X₇₆₁₀ + X₁₁₁₆₅, \(5\) X₇₆₁₉ - X₇₇₆₄, \(4\) X₇₆₁₉ + X₇₇₈₀, \(3\) X₇₆₂₂ - X₁₁₁₆₅, \(4\) X₇₇₆₄ + \(5\) X₇₇₈₀, X₇₇₆₄ + \(5\) X₃₄₅₀₆, X₇₇₈₀ - \(4\) X₃₄₅₀₆, \(3\) X₈₁₇₆ - X₂₃₃₃₄, \(13\) X₈₁₇₆ - \(3\) X₄₄₆₇₈, \(3\) X₈₁₈₂ + X₂₃₃₃₄, \(13\) X₈₁₈₂ + \(3\) X₄₄₆₇₈, X₉₇₇₀ - \(9\) X₁₅₇₀₉, X₉₇₇₁ - \(3\) X₁₁₅₃₉, \(13\) X₁₀₃₀₃ - X₃₄₅₁₁, X₁₁₁₄₈ - \(33\) X₁₅₇₂₁, X₁₁₁₈₄ - \(5\) X₁₅₆₉₄, X₁₂₀₄₀ - \(5\) X₁₅₇₁₃, X₁₃₄₆₈ + \(5\) X₁₅₇₁₃, \(3\) X₁₄₁₆₁ - \(2\) X₁₄₁₆₂, \(3\) X₁₅₅₉₇ - X₁₆₅₀₉, \(5\) X₁₅₆₉₃ + X₁₈₅₄₆, \(7\) X₁₅₇₀₁ + X₄₀₇₂₇, \(11\) X₁₅₇₂₀ + X₃₄₅₀₅, \(13\) X₂₃₃₃₄ - \(9\) X₄₄₆₇₈, X₂₅₄₈₆ - \(5\) X₃₁₂₇₄

• X₁₁₅₃ = \(8 G + 3 O + K + Csc[w]^{2} (G - K)\)

X₁₁₅₃ lies on the cubic K1258.

▸ Incident Central Lines72 lines

{2, 187}, {3, 7617}, {30, 32414}, {39, 8859}, {140, 524}, {141, 8787}, {404, 7621}, {511, 7606}, {538, 5054}, {543, 549}, {574, 8860}, {599, 10485}, {618, 33475}, {619, 33474}, {620, 11168}, {631, 5485}, {632, 7843}, {671, 8589}, {754, 9771}, {2482, 37688}, {2549, 23053}, {3054, 5461}, {3523, 7620}, {3524, 7615}, {3526, 7775}, {3788, 21356}, {4045, 44401}, {5032, 31401}, {5092, 9830}, {6683, 13330}, {6719, 10354}, {7496, 42008}, {7749, 7817}, {7769, 41136}, {7810, 41133}, {7815, 21358}, {7816, 33274}, {7848, 22110}, {7849, 33000}, {7854, 33204}, {7861, 33215}, {7880, 9167}, {7883, 16923}, {7886, 8359}, {8356, 14971}, {8587, 10302}, {8588, 11317}, {8597, 39601}, {8703, 20112}, {9166, 33273}, {9466, 11152}, {9770, 15709}, {9877, 37455}, {10303, 34511}, {11147, 11151}, {11148, 15721}, {11164, 32456}, {11184, 11842}, {11645, 40278}, {12040, 13468}, {14159, 14160}, {14161, 14162}, {15513, 33013}, {15693, 18546}, {15701, 40727}, {15720, 34505}, {19911, 21163}, {25486, 31274}, {32480, 37512}, {33476, 35304}, {33477, 35303}, {41895, 43619}

▸ Functional Relations4 relations

• X₁₁₅₃ = midpoint of X_i and X_j for these {i,j}: {2, 5569}, {3, 7617}, {549, 15597}, {7610, 7622}, {7619, 34506}, {7620, 34504}, {8176, 8182}, {8703, 20112}, {12040, 13468}

• X₁₁₅₃ = reflection of X_i in X_j for these {i,j}: {7619, 140}, {14160, 14159}

• X₁₁₅₃ = complement of X₈₁₇₆

• X₁₁₅₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 7771, 31173}, {2, 8182, 8176}, {5054, 7610, 7622}, {5569, 8176, 8182}

Notes

Floor van Lamoen Problem 10830, American Mathematical Monthly 107 (2000) 863; solution by the editors, 109 (2002) 396-397.

Numerous messages about this circle and its center can be accessed from the Hyacinthos archive using “Floor’s Monthly problem” as search words. M. Stevanovic’s message (#5599, 5/28/02) gives coordinates.