X₃₁

2nd POWER POINT

X₃₁ = barycentric product of PU(8)

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

X₃₁ (2nd POWER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {6, 42}

• Related Centers: X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₆ (SYMMEDIAN POINT), X₄₂ (CROSSPOINT OF INCENTER AND SYMMEDIAN POINT)

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

Barycentrics: \(a^{3} : b^{3} : c^{3}\)

Search Value (6, 9, 13): \(0.542275931742\)

▸ Alternate Coordinates4 forms

• Trilinears: \(1 - \cos 2A : 1 - \cos 2B : 1 - \cos 2C\)

• Trilinears: \(\cot B + \cot C : :\)

• Trilinears: \(S_{B} + S_{C} : :\)

• Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = distance between A and de Longchamps line

▸ Linear Combinations1 combination

• X₃₁ = \((r^{2} + s^{2})\) X₁ - \(6rR\) X₂ - \(4r^{2}\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines123 lines

{1, 21}, {2, 171}, {3, 601}, {4, 3072}, {6, 42}, {8, 987}, {9, 612}, {10, 964}, {19, 204}, {25, 608}, {28, 2282}, {32, 41}, {33, 2250}, {34, 1254}, {35, 386}, {36, 995}, {37, 2214}, {40, 580}, {43, 100}, {44, 210}, {48, 560}, {51, 181}, {56, 154}, {57, 105}, {65, 1104}, {72, 976}, {75, 82}, {76, 734}, {86, 2296}, {91, 1087}, {92, 162}, {99, 715}, {101, 609}, {106, 2163}, {110, 593}, {112, 2249}, {158, 2190}, {163, 923}, {165, 2999}, {172, 1613}, {184, 604}, {197, 2183}, {198, 2255}, {199, 2277}, {200, 1261}, {218, 1260}, {222, 1458}, {226, 3011}, {237, 904}, {240, 1748}, {278, 1430}, {284, 2258}, {292, 1915}, {354, 1279}, {388, 1935}, {404, 978}, {497, 1936}, {516, 1754}, {561, 722}, {582, 3579}, {607, 2357}, {649, 884}, {663, 2423}, {669, 875}, {678, 3158}, {692, 2877}, {701, 789}, {708, 1502}, {740, 3187}, {743, 825}, {745, 827}, {759, 994}, {775, 1097}, {872, 2220}, {893, 1691}, {899, 1376}, {901, 2382}, {937, 1103}, {940, 1001}, {982, 3218}, {984, 3219}, {990, 1709}, {999, 1149}, {1066, 3157}, {1098, 2363}, {1124, 3076}, {1182, 3192}, {1210, 1771}, {1331, 2991}, {1335, 3077}, {1393, 1454}, {1403, 1428}, {1427, 1456}, {1438, 2279}, {1450, 1470}, {1474, 2215}, {1486, 2260}, {1572, 2170}, {1582, 1740}, {1616, 3304}, {1633, 3123}, {1820, 1953}, {1836, 3120}, {1910, 2186}, {1911, 1922}, {1917, 2085}, {1927, 1967}, {1932, 1973}, {1951, 3010}, {1974, 2281}, {1979, 2107}, {2003, 2078}, {2054, 2248}, {2083, 2156}, {2153, 2154}, {2188, 2638}, {2242, 3230}, {2264, 3198}, {2274, 3286}, {2318, 2911}, {3074, 3085}, {3075, 3086}, {3220, 3415}

▸ Functional Relations45 relations

• X₃₁ = isogonal conjugate of X₇₅

• X₃₁ = isotomic conjugate of X₅₆₁

• X₃₁ = complement of X₆₃₂₇

• X₃₁ = anticomplement of X₂₈₈₇

• X₃₁ = anticomplementary conjugate of anticomplement of X₃₈₈₁₃

• X₃₁ = circumcircle-inverse of X₅₁₆₁

• X₃₁ = X_i-Ceva conjugate of X_j for these (i,j): (1, 48), (6, 41), (9, 205), (58, 6), (82, 1)

• X₃₁ = X₂₁₃-cross conjugate of X₆

• X₃₁ = crosspoint of X_i and X_j for these (i,j): (1, 19), (6, 56)

• X₃₁ = crosssum of X_i and X_j for these (i,j): (1, 63), (2, 8), (7, 347), (10, 321), (239, 1281), (244, 514), (307, 1441), (523, 1086), (693, 1111)

• X₃₁ = crossdifference of every pair of points on line X₅₁₄X₆₆₁

• X₃₁ = X₁₄₀₃-Hirst inverse of X₁₄₂₈

• X₃₁ = X_i-aleph conjugate of X_j for these (i,j): (82, 31), (83, 75)

• X₃₁ = X_i-beth conjugate of X_j for these (i,j): (21, 993), (55, 55), (109, 31), (110, 57), (643, 31), (692, 31)

• X₃₁ = barycentric product of PU(8)

• X₃₁ = vertex conjugate of PU(8)

• X₃₁ = bicentric sum of PU(i) for these i: 23, 48

• X₃₁ = PU(23)-harmonic conjugate of X₆₆₁

• X₃₁ = PU(48)-harmonic conjugate of X₆₄₉

• X₃₁ = trilinear product of PU(36)

• X₃₁ = trilinear product X₅₅*X₅₆

• X₃₁ = trilinear pole of line X₆₆₇X₇₈₈

• X₃₁ = pole wrt polar circle of trilinear polar of X₁₉₆₉

• X₃₁ = X₄₈-isoconjugate (polar conjugate) of X₁₉₆₉

• X₃₁ = X₆-isoconjugate of X₇₆

• X₃₁ = X₉₂-isoconjugate of X₆₃

• X₃₁ = trilinear square of X₆

• X₃₁ = trilinear cube root of X₁₉₁₇

• X₃₁ = vertex conjugate of foci of incentral inellipse

• X₃₁ = perspector of ABC and extraversion triangle of X₃₁ (which is also the anticevian triangle of X₃₁)

• X₃₁ = {X₁,X₁₇₀₇}-harmonic conjugate of X₆₃

• X₃₁ = perspector of ABC and unary cofactor triangle of extraversion triangle of X₇

• X₃₁ = perspector of ABC and unary cofactor triangle of extraversion triangle of X₈ (2nd Conway triangle)

• X₃₁ = perspector of ABC and unary cofactor triangle of 4th Conway triangle

• X₃₁ = perspector of unary cofactor triangles of 2nd and 4th Conway triangles

• X₃₁ = perspector of unary cofactor triangles of Gemini triangles 2 and 30

• X₃₁ = perspector of ABC and cross-triangle of Gemini triangles 33 and 34

• X₃₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 33

• X₃₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 34

• X₃₁ = barycentric product of vertices of Gemini triangle 33

• X₃₁ = barycentric product of vertices of Gemini triangle 34

• X₃₁ = barycentric product of (nonreal) circumcircle intercepts of the antiorthic axis

• X₃₁ = center of circumconic locus of trilinear poles of lines passing through X₃₂₆₆₄

• X₃₁ = perspector of circumconic centered at X₃₂₆₆₄

• X₃₁ = X₂-Ceva conjugate of X₃₂₆₆₄

Notes

Let A'B'C' be the circumsymmedial triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. Then A″, B″, C″ are collinear on line X₆₆₇X₇₈₈ (the trilinear polar of X₃₁). The lines AA″, BB″, CC″ concur in X₃₁. (Randy Hutson, February 10, 2016)

Let A'B'C' be the Apus triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₁. (Randy Hutson, February 10, 2016)

Let A'B'C' be the Ara triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₁. (Randy Hutson, February 10, 2016)

Define the 1st Kenmotu diagonals triangle as the triangle formed by the diagonals of the squares in the Kenmotu configuration with center X₃₇₁ that do not include X₃₇₁. Define the 2nd Kenmotu diagonals triangle as the triangle formed by the diagonals of the squares in the Kenmotu configuration with center X₃₇₂ that do not include X₃₇₂. (Randy Hutson, February 10, 2016)

Let A₁B₁C₁ and A₂B₂C₂ be the 1st and 2nd Kenmotu diagonals triangles. Let A' be the trilinear product A₁*A₂, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₁. (Randy Hutson, February 10, 2016)

X₃₁ is the Brianchon point (perspector) of the inellipse that is the trilinear square of the Lemoine axis. The center of the inellipse is X₁₆₅₈₄. (Randy Hutson, October 15, 2018)

X₃₁ is the {X₁,X₆₃}-harmonic conjugate of X₃₈.