X₂₄₈₂

STEINER-INELLIPSE-ANTIPODE OF X₁₁₅

X₂₄₈₂ is the Steiner-inellipse-antipode of X₁₁₅.

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

X₂₄₈₂ (STEINER-INELLIPSE-ANTIPODE OF X₁₁₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 99}

• Related Centers: X₂ (CENTROID), X₉₉ (STEINER POINT)

Trilinears: \(bc(2a^{2} - b^{2} - c^{2})^{2} : :\)

Search Value (6, 9, 13): \(3.034049123068\)

X₂₄₈₂ lies on the Steiner inscribed ellipse, the bicevian conic of X₂ and X₉₉.

▸ Incident Central Lines12 lines

{2, 99}, {3, 67}, {30, 114}, {32, 1992}, {39, 597}, {187, 524}, {230, 5215}, {351, 690}, {530, 619}, {531, 618}, {538, 1569}, {1086, 1125}

▸ Functional Relations21 relations

• X₂₄₈₂ = midpoint of X₂ and X₉₉

• X₂₄₈₂ = reflection of X_i in X_j for these (i,j): (2, 620), (115, 2)

• X₂₄₈₂ = isogonal conjugate of X₁₀₆₃₀

• X₂₄₈₂ = complement of X₆₇₁

• X₂₄₈₂ = complementary conjugate of X₆₂₅

• X₂₄₈₂ = X_i-Ceva conjugate of X_j for these (i,j): (2, 524), (88, 690)

• X₂₄₈₂ = crosspoint of X₂ and X₅₂₄

• X₂₄₈₂ = crosssum of X₆ and X₁₁₁

• X₂₄₈₂ = crosssum of circumcircle intercepts of Schoute circle

• X₂₄₈₂ = projection from Steiner circumellipse to Steiner inellipse of X₉₉

• X₂₄₈₂ = perspector of circumconic centered at X₅₂₄

• X₂₄₈₂ = center of circumconic that is locus of trilinear poles of lines passing through X₅₂₄ (hyperbola {A, B, C,X₂,X₉₉})

• X₂₄₈₂ = crossdifference of every pair of points on line X₁₁₁X₃₅₁

• X₂₄₈₂ = isogonal conjugate wrt medial triangle of X₆₂₅

• X₂₄₈₂ = barycentric square of X₅₂₄

• X₂₄₈₂ = midpoint of X₅₄₆₃ and X₅₄₆₄

• X₂₄₈₂ = X₅ of anti-McCay triangle

• X₂₄₈₂ = antipode of X₂ in the bicevian conic of X₂ and X₉₉

• X₂₄₈₂ = centroid of mid-triangle of antipedal triangles of X₁₃ and X₁₄

• X₂₄₈₂ = X₂-of-X₁₈₇-adjunct-anti-altimedial-triangle

• X₂₄₈₂ = QA-P₃ (Gergonne-Steiner Point) of quadrangle ABCX₂ (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/18-qa-p3.html)

Notes

X₂₄₈₂ is the midpoint of the centers of the (equilateral) antipedal triangles of X₁₃ and X₁₄. (Randy Hutson, 9/23/2011)

Let H_aH_bH_c be the orthic triangle. Then X₂₄₈₂ is the radical center of the Schoute circles of triangles AH_bH_c, BH_cH_a, CH_aH_b. (Randy Hutson, July 31 2018)