X₉₄₆

MIDPOINT OF X₁ AND X₄

X₉₄₆ is the midpoint of X₁ and X₄.

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

X₉₄₆ (MIDPOINT OF X₁ AND X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}

• Related Centers: X₁ (INCENTER), X₄ (ORTHOCENTER)

Trilinears: \(bc[a^{3}(b + c) + (b - c)^{2}(a^{2} - ab - ac - b^{2} - c^{2} - 2bc)] : :\)

Search Value (6, 9, 13): \(-1.993155450735\)

▸ Alternate Coordinates2 forms

• Trilinears: \(r + 2 R \cos B \cos C : :\)

• Trilinears: \(2 \tan A + \tan B + \tan C + (\sec A)(\sin B + \sin C) : :\)

▸ Linear Combinations1 combination

• X₉₄₆ = X₁ + X₄

▸ Incident Central Lines19 lines

{1, 4}, {2, 40}, {3, 142}, {5, 10}, {7, 84}, {8, 908}, {11, 65}, {29, 102}, {30, 551}, {46, 499}, {56, 1012}, {79, 104}, {165, 631}, {238, 580}, {355, 381}, {392, 442}, {496, 942}, {546, 952}, {951, 1067}

▸ Functional Relations17 relations

• X₉₄₆ = midpoint of X_i and X_j for these (i,j): (1, 4), (40, 962)

• X₉₄₆ = reflection of X_i in X_j for these (i,j): (3, 1125), (10, 5)

• X₉₄₆ = isogonal conjugate of X₉₄₇

• X₉₄₆ = complement of X₄₀

• X₉₄₆ = incircle-inverse of X₁₇₈₅

• X₉₄₆ = crosspoint of X_i and X_j for these (i,j): (2, 309), (7, 92)

• X₉₄₆ = crosssum of X₄₈ and X₅₅

• X₉₄₆ = incenter-of-Euler-triangle

• X₉₄₆ = X₄-of-3rd-Euler-triangle

• X₉₄₆ = X₂₀-of-4th-Euler-triangle

• X₉₄₆ = X₁₁₃-of-Fuhrmann-triangle

• X₉₄₆ = Johnson-isogonal conjugate of X₃₇₈₂₃

• X₉₄₆ = center of conic that is the locus of orthopoles of lines passing through X₁

• X₉₄₆ = radical center of circles centered at excenters and internally tangent to nine-point circle

• X₉₄₆ = harmonic center of 1st & 2nd Johnson-Yff circles

• X₉₄₆ = X₅₄₄₆-of-excentral-triangle

• X₉₄₆ = Johnson-to-Ehrmann-mid similarity image of X₁

Notes

Let A' be the midpoint of BC, and define B' and C' cyclically. A″ be the midpoint of AX₁, and define B″ and C″ cyclically. Let A″’B″’C″’ be the incentral triangle. Let A* be the orthocenter of A’A″A″’, and define B* and C* cyclically. The lines A″A*, B″B*, C″C* concur in X₉₄₆. (Randy Hutson, November 18, 2015)

Let A' be the intersection of these three lines: the perpendicular from midpoint of segment CA to line BX₁, the perpendicular from midpoint of segment AB to line CX₁, and the perpendicular from midpoint of segment AX₁ to line BC. Define B' and C' cyclically. The circumcenter of A'B'C' is X₉₄₆. Note that A'B'C' is the complement of the excentral triangle; also A'B'C' is the extraversion triangle of X₁₀. (Randy Hutson, November 18, 2015)

Let A' be the midpoint of A and X₁, and define B' and C' cyclically. The orthocenter of A'B'C' is X₉₄₆. (Randy Hutson, November 18, 2015)