X₁₅₁₁

FERMAT CROSSSUM

X₁₅₁₁ = complement of X₂₆₅

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

X₁₅₁₁ (FERMAT CROSSSUM) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 74}, {30, 113}

• Related Centers: X₃ (CIRCUMCENTER), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₃₀ (EULER INFINITY POINT), X₁₁₃ (JERABEK ANTIPODE)

Trilinears: \(\csc(B + \pi /3)\csc(C - \pi /3) + \csc(C + \pi /3)\csc(B - \pi /3)\)

Barycentrics: \(a^{2}[(a^{2} - b^{2} - c^{2})^{2} - b^{2}c^{2}][2a^{4} - a^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}] : :\)

Search Value (6, 9, 13): \(3.844962934149\)

▸ Alternate Coordinates1 form

• Trilinears: \((\sin^{2} A - 3 \cos^{2} A)(\sin B \sin C - 3 \cos B \cos C)\)

X₁₅₁₁ lies on (Johnson circumconic of medial triangle), the bicevian conic of X₂ and X₁₁₀.

▸ Incident Central Lines13 lines

{2, 265}, {3, 74}, {24, 1112}, {30, 113}, {36, 1464}, {125, 128}, {141, 542}, {146, 376}, {184, 974}, {186, 323}, {214, 960}, {249, 842}, {389, 1493}

▸ Functional Relations18 relations

• X₁₅₁₁ = midpoint of X_i and X_j for these (i,j): (3, 110), (74, 399)

• X₁₅₁₁ = reflection of X_i in X_j for these (i,j): (125, 140), (1539, 113)

• X₁₅₁₁ = complement of X₂₆₅

• X₁₅₁₁ = complementary conjugate of X₂₀₇₂

• X₁₅₁₁ = isotomic conjugate of polar conjugate of X₃₉₁₇₆

• X₁₅₁₁ = crosssum of circumcircle-intercepts of line PU(174) (line X₄X₅₂₃)

• X₁₅₁₁ = X_i-Ceva conjugate of X_j for these (i,j): (2, 3284), (3, 1154), (110, 526)

• X₁₅₁₁ = crosspoint of X_i and X_j for these (i,j): (2, 340), (15, 16)

• X₁₅₁₁ = crosssum of X₁₃ and X₁₄

• X₁₅₁₁ = crossdifference of every pair of points on the line X₁₆₃₇X₁₉₈₉

• X₁₅₁₁ = inverse-in-circumcircle of X₃₉₉

• X₁₅₁₁ = perspector of the circumconic centered at X₃₂₈₄

• X₁₅₁₁ = {X₃,X₃₉₉}-harmonic conjugate of X₇₄

• X₁₅₁₁ = orthocentroidal-to-ABC similarity image of X₅

• X₁₅₁₁ = 4th-Brocard-to-circumsymmedial similarity image of X₅

• X₁₅₁₁ = reflection of X₅ in X₅₉₇₂

• X₁₅₁₁ = antipode of X₅ in the bicevian conic of X₂ and X₁₁₀

• X₁₅₁₁ = X₁₂₅₁₅-of-orthic-triangle if ABC is acute

Notes

Let A'B'C' be the cevian triangle of X₃₀. Let A″, B″, C″ be the inverse-in-circumcircle of A', B', C'. The lines AA″, BB″, CC″ concur in X₁₅₁₁. Let N_A be the reflection of X₅ in the perpendicular bisector of segment BC, and define N_B and N_C cyclically; then X₁₅₁₁ = X₁₈₆-of-N_AN_BN_C. (Randy Hutson, August 26, 2014)

Let M_aM_bM_c be the medial triangle. Let L be the line through M_a parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L',M',N' concur in X₁₅₁₁. Let (O') be the circle passing through the points X₃, X₄, X₃₉₉, X₆₀₆₉, let Q be the radical axis (O') and the circumcircle; then X₁₅₁₁ = Q∩X₃X₇₄. (Randy Hutson, August 26, 2014)

Let U be the Simson line of X₁₁₀, which is the line X₃₀X₁₁₃. Let V be the line normal to the circumcircle at X₁₁₀, which is the line X₃X₇₄. Then X₁₅₁₁ = U∩V. (Randy Hutson, January 29, 2015)