X₃₀₉₁

INTERSECTION OF LINES X₂X_3 AND X₁₁X_153

X₃₀₉₁ = midpoint of X₄ and X₆₃₁

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

X₃₀₉₁ (INTERSECTION OF LINES X₂X_3 AND X₁₁X_153) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 631}, {2, 3} (Euler line)

• Related Centers: X₄ (ORTHOCENTER), X₆₃₁ (3/5*OG), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(\cos B \cos C + \cos(B - C) : :\)

Barycentrics: \(S^{2} + 2 S_{B} S_{C} : :\)

Search Value (6, 9, 13): \(-0.693026488877\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a^{4} + 2 a^{2} (b^{2} + c^{2}) - 3 (b^{2} - c^{2})^{2} : :\)

▸ Incident Central Lines49 lines

{2, 3}, {7, 1210}, {8, 908}, {10, 962}, {11, 153}, {12, 497}, {68, 1173}, {110, 578}, {114, 148}, {115, 147}, {116, 152}, {118, 150}, {119, 149}, {124, 151}, {125, 146}, {145, 355}, {155, 1994}, {156, 567}, {193, 576}, {194, 262}, {226, 938}, {253, 264}, {324, 1093}, {347, 1838}, {390, 1479}, {485, 1132}, {486, 1131}, {495, 1058}, {496, 1056}, {516, 1698}, {569, 1614}, {637, 1271}, {638, 1270}, {1007, 1975}, {1329, 2550}, {1348, 2543}, {1349, 2542}, {1454, 1776}, {1478, 3086}, {1506, 2549}, {1676, 2547}, {1677, 2546}, {1788, 1836}, {1853, 2883}, {2009, 2545}, {2010, 2544}, {2551, 2886}, {3068, 3071}, {3069, 3070}

▸ Functional Relations16 relations

• X₃₀₉₁ = midpoint of X₄ and X₆₃₁

• X₃₀₉₁ = reflection of X_i in X_j for these (i,j): (3, 632), (631, 1656), (1656, 5), (3522, 631)

• X₃₀₉₁ = inverse-in-orthocentroidal-circle of X₂₀

• X₃₀₉₁ = complement of X₃₅₂₂

• X₃₀₉₁ = anticomplement of X₆₃₁

• X₃₀₉₁ = circumcircle-inverse of X₃₇₉₄₀

• X₃₀₉₁ = homothetic center of orthocentroidal triangle and X₄-Brocard triangle

• X₃₀₉₁ = homothetic center of anticomplementary triangle and cross-triangle of ABC and Euler triangle

• X₃₀₉₁ = homothetic center of Euler triangle and cross-triangle of ABC and Euler triangle

• X₃₀₉₁ = circumcenter of cross-triangle of Euler and anti-Euler triangles

• X₃₀₉₁ = homothetic center of X₂-altimedial and X₄-anti-altimedial triangles

• X₃₀₉₁ = homothetic center of X₂₀-altimedial and X₂₀-anti-altimedial triangles

• X₃₀₉₁ = homothetic center of X₂₀-anti-altimedial and orthocentroidal triangles

• X₃₀₉₁ = X₇₈₉₇-of-orthic-triangle if ABC is acute

• X₃₀₉₁ = homothetic center of Ehrmann mid-triangle and anti-Euler triangle

• X₃₀₉₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (3, 5, 3090), (4, 5, 2), (4, 3544, 3090)

Notes

As a point on the Euler line, X₃₀₉₁ has Shinagawa coefficients \((1,2)\).

Let Eu_aEu_bEu_c be the Euler triangle. Let A″ be the cevapoint of Eu_b and Eu_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₀₉₁. (Randy Hutson, December 10, 2016)

Let Eu_aEu_bEu_c be the Euler triangle. Let A* be the centroid of AEu_bEu_c, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₃₀₉₁. (Randy Hutson, December 10, 2016)