X₁₁₆

MIDPOINT OF X₄ AND X₁₀₃

X₁₁₆ is the midpoint of X₄ and X₁₀₃.

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

X₁₁₆ (MIDPOINT OF X₄ AND X₁₀₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 103}

• Related Centers: X₄ (ORTHOCENTER), X₁₀₃ (ANTIPODE OF X₁₀₁)

Trilinears: \(bc[(b - c)^{2}(b^{2} + bc + c^{2} - ab - ac)] : :\)

Barycentrics: \((b - c)^{2}(b^{2} + bc + c^{2} - ab - ac) : :\)

Search Value (6, 9, 13): \(3.283799820557\)

X₁₁₆ lies on the nine-point circle.

▸ Incident Central Lines8 lines

{2, 101}, {4, 103}, {5, 118}, {10, 120}, {115, 1086}, {119, 142}, {121, 141}, {124, 928}

▸ Functional Relations13 relations

• X₁₁₆ = midpoint of X_i and X_j for these (i,j): (4, 103), (101, 150)

• X₁₁₆ = reflection of X₁₁₈ in X₅

• X₁₁₆ = isotomic conjugate of isogonal conjugate of X₂₀₉₇₄

• X₁₁₆ = complement of X₁₀₁

• X₁₁₆ = complementary conjugate of X₅₁₄

• X₁₁₆ = X₄-Ceva conjugate of X₅₁₄

• X₁₁₆ = polar conjugate of isogonal conjugate of X₂₂₀₈₄

• X₁₁₆ = inverse-in-excircles-radical-circle of X₃₀₃₄

• X₁₁₆ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₆₇₅

• X₁₁₆ = anticenter of cyclic quadrilateral ABCX₁₀₃

• X₁₁₆ = Λ(Gergonne line), wrt orthic triangle

• X₁₁₆ = X₂-Ceva conjugate of X₆₅₈₆

• X₁₁₆ = X₁₀₁-of-medial triangle.

Notes

Let H_aH_bH_c be the orthic triangle. Let L_a be the Soddy line of triangle AH_bH_c, and define L_b and L_c cyclically. Let A″ = L_b∩L_c, and define B″ and C″ cyclically. The lines H_aA″, H_bB″, H_cC″ concur in X₁₁₆, which is X₃-of-A″B″C″. (Randy Hutson, July 31 2018)