X₁₆₆₀

1st GRINBERG MIDPOINTS PERSPECTOR

The lines AX_A, BX_B, CX_C concur in X₂₀.

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

X₁₆₆₀ (1st GRINBERG MIDPOINTS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {394, 1619}

• Related Centers: X₃₉₄ (-CEVA CONJUGATE OF X₃), X₁₆₁₉ (TCC-PERSPECTOR OF X₆₃)

Trilinears: \(a^{3}[b^{8} + c^{8} - a^{8} - 2a^{2}(b^{6} + c^{6}) + 2b^{2}c^{2}(b^{4} + c^{4} - 5a^{4} + 3a^{2}b^{2} + 3a^{2}c^{2} - 3b^{2}c^{2}) + 2a^{6}(b^{2} + c^{2})]\)

Search Value (6, 9, 13): \(-1.975733876144\)

▸ Incident Central Lines7 lines

{6, 25}, {30, 156}, {110, 1370}, {394, 1619}, {578, 1596}, {1092, 1498}, {1368, 1503}

▸ Functional Relations3 relations

• X₁₆₆₀ = midpoint of X₃₉₄ and X₁₆₁₉

• X₁₆₆₀ = X₂₀-Ceva conjugate of X₅₇₇

• X₁₆₆₀ = X₂₅-of-A'B'C', as described by Tran Quang Huyng, ADGEOM #2697 (8/26/2015)

Notes

Let A_B be the point in which the line through A perpendicular to CA meets line BC, and define points A_C, B_C, B_A, C_A, C_B functionally. Let

X_A = midpoint{A_B, A_C},

• Y_A = midpoint{B_A, C_A},

• Z_A = midpoint{B_C, C_B},

and define X_B, X_C, Y_B, Y_C, Z_B, Z_C functionally.

The lines AY_A, BY_B, CY_C concur in X₃₉₃.

The lines AZ_A, BZ_B, CZ_C concur in X₆.

The lines X_AY_A, X_BY_B, X_CY_C concur in X₁₆₆₀.

The lines Y_AZ_A, Y_BZ_B, Y_CZ_C concur in X₃.

The lines Z_AX_A, Z_BX_B, Z_CX_C concur in X₁₆₆₁.

Contributed by Darij Grinberg, August 24, 2003; see Hyacinthos #7225.