X₃₀

EULER INFINITY POINT

X₃₀ is the point of intersection of the Euler line and the line at infinity. Thus, each of the lines listed below is parallel to the Euler line.

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

X₃₀ (EULER INFINITY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {13, 15}

• Related Centers: X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₃ (1st ISOGONIC CENTER), X₁₅ (1st ISODYNAMIC POINT)

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

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = 2a^{4} - (b^{2} - c^{2})^{2} - a^{2}(b^{2} + c^{2})\)

▸ Alternate Coordinates4 forms

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

• Trilinears: \(2 \sec A - \sec B \sec C : :\)

• Trilinears: \(\sin B \sin C - 3 \cos B \cos C : :\)

• Barycentrics: \(S^{2} - 3 S_{B} S_{C} : :\)

▸ Linear Combinations1 combination

• X₃₀ = X₂ - X₃; if (i) and X_j are on the Euler line, then X₃₀ = X_i - X_j

X₃₀ lies on the Neuberg cubic, the Darboux quintic.

▸ Incident Central Lines244 lines

{1, 79}, {2, 3}, {6, 2549}, {7, 3488}, {8, 3578}, {9, 3587}, {10, 3579}, {11, 36}, {12, 35}, {13, 15}, {14, 16}, {17, 5238}, {18, 5237}, {32, 5254}, {33, 1060}, {34, 1062}, {40, 191}, {46, 1837}, {49, 1614}, {50, 1989}, {51, 5946}, {52, 185}, {53, 577}, {54, 3521}, {55, 495}, {56, 496}, {57, 3586}, {58, 1834}, {61, 397}, {62, 398}, {63, 3419}, {64, 68}, {65, 1770}, {69, 3426}, {74, 265}, {80, 484}, {84, 3928}, {98, 671}, {99, 316}, {100, 2687}, {101, 2688}, {102, 2689}, {103, 2690}, {104, 1290}, {105, 2691}, {106, 2692}, {107, 2693}, {108, 2694}, {109, 2695}, {110, 477}, {111, 2696}, {112, 2697}, {113, 1495}, {114, 2482}, {115, 187}, {119, 2077}, {128, 6592}, {133, 3184}, {137, 6150}, {141, 3098}, {143, 389}, {146, 323}, {148, 385}, {154, 5654}, {155, 1498}, {156, 1147}, {165, 5587}, {182, 597}, {226, 4304}, {250, 6530}, {262, 598}, {284, 1901}, {298, 616}, {299, 617}, {315, 1975}, {329, 3940}, {340, 1494}, {371, 3070}, {372, 3071}, {388, 3295}, {390, 1056}, {485, 1151}, {486, 1152}, {489, 638}, {490, 637}, {497, 999}, {498, 5217}, {499, 5204}, {511, 512}, {551, 946}, {553, 942}, {567, 5012}, {568, 3060}, {574, 3815}, {582, 1724}, {590, 6200}, {599, 1350}, {615, 6396}, {618, 623}, {619, 624}, {620, 625}, {664, 5195}, {841, 1302}, {908, 5440}, {910, 5179}, {925, 5962}, {935, 1297}, {938, 5708}, {944, 962}, {956, 3434}, {993, 2886}, {1043, 1330}, {1058, 3600}, {1117, 5671}, {1125, 3824}, {1131, 6407}, {1132, 6408}, {1141, 1157}, {1145, 5176}, {1146, 5011}, {1155, 1737}, {1160, 5860}, {1161, 5861}, {1213, 4877}, {1216, 5907}, {1285, 5304}, {1292, 2752}, {1293, 2758}, {1294, 1304}, {1295, 2766}, {1296, 2770}, {1319, 1387}, {1337, 3479}, {1338, 3480}, {1351, 1353}, {1376, 3820}, {1465, 1877}, {1490, 5763}, {1565, 4872}, {1587, 3311}, {1588, 3312}, {1625, 3289}, {1691, 6034}, {1699, 3576}, {1750, 5720}, {1754, 5398}, {1765, 5755}, {1768, 5535}, {1807, 3465}, {1838, 1852}, {1865, 2193}, {1870, 3100}, {1990, 3163}, {2021, 2023}, {2093, 5727}, {2094, 2095}, {2132, 2133}, {2292, 5492}, {2456, 5182}, {2548, 5013}, {2646, 4870}, {2654, 4303}, {2895, 4720}, {2931, 2935}, {2968, 5081}, {3003, 6128}, {3023, 6023}, {3027, 6027}, {3035, 3814}, {3053, 3767}, {3068, 6221}, {3069, 6398}, {3085, 5229}, {3086, 5225}, {3167, 5656}, {3255, 3577}, {3260, 6148}, {3292, 5609}, {3303, 4309}, {3304, 4317}, {3357, 5894}, {3424, 5485}, {3429, 4052}, {3436, 5687}, {3481, 3482}, {3485, 4305}, {3486, 4295}, {3487, 4313}, {3565, 5203}, {3589, 4045}, {3665, 4056}, {3703, 4680}, {3746, 4330}, {3829, 5450}, {3911, 5122}, {3917, 5891}, {3925, 5251}, {4030, 4692}, {4252, 5292}, {4296, 6198}, {4298, 5045}, {4301, 5882}, {4325, 4857}, {4421, 6256}, {4424, 5724}, {4511, 5057}, {4669, 5493}, {4677, 5881}, {4999, 5267}, {5008, 5355}, {5010, 5432}, {5032, 5093}, {5103, 5149}, {5107, 5477}, {5119, 5252}, {5180, 6224}, {5188, 6248}, {5207, 6393}, {5418, 6409}, {5420, 6410}, {5424, 5561}, {5448, 5893}, {5459, 5478}, {5460, 5479}, {5461, 6036}, {5463, 5473}, {5464, 5474}, {5538, 6326}, {5562, 5876}, {5603, 5731}, {5657, 5790}, {5703, 5714}, {5732, 5805}, {5758, 6223}, {5759, 5779}, {5858, 5864}, {5859, 5865}, {5889, 6241}, {5892, 5943}, {6104, 6107}, {6105, 6106}, {6193, 6225}, {6237, 6254}, {6238, 6285}

▸ Functional Relations34 relations

• X₃₀ = isogonal conjugate of X₇₄

• X₃₀ = isotomic conjugate of X₁₄₉₄

• X₃₀ = anticomplementary conjugate of X₁₄₆

• X₃₀ = complementary conjugate of X₁₁₃

• X₃₀ = orthopoint of X₅₂₃

• X₃₀ = X_i-Ceva conjugate of X_j for these (i,j): (4, 113), (265, 5), (476, 523)

• X₃₀ = cevapoint of X₃ and X₃₉₉

• X₃₀ = crosspoint of X_i and X_j for these (i,j): (13, 14), (94, 264)

• X₃₀ = crosssum of X_i and X_j for these (i,j): (15, 16), (50, 184)

• X₃₀ = crossdifference of every pair of points on line X₆X₆₄₇

• X₃₀ = ideal point of PU(30)

• X₃₀ = vertex conjugate of PU(87)

• X₃₀ = perspector of circumconic centered at X₃₁₆₃

• X₃₀ = center of circumconic that is locus of trilinear poles of lines passing through X₃₁₆₃

• X₃₀ = X₂-Ceva conjugate of X₃₁₆₃

• X₃₀ = trilinear pole of line X₁₆₃₆X₁₆₃₇ (the line that is the tripolar centroid of the Euler line)

• X₃₀ = X₅₁₇-of-orthic triangle if ABC is acute

• X₃₀ = X₅₄₂-of-1st Brocard triangle

• X₃₀ = crosspoint of X₃ and X₃₉₉ wrt both the excentral and tangential triangles

• X₃₀ = crosspoint of X₆₁₆ and X₆₁₇ wrt both the excentral and anticomplementary triangles

• X₃₀ = cevapoint of X₆₁₆ and X₆₁₇

• X₃₀ = X₆-isoconjugate of X₂₃₄₉

• X₃₀ = perspector of 2nd isogonal triangle of X₄ and cross-triangle of ABC and 2nd isogonal triangle of X₄

• X₃₀ = Thomson isogonal conjugate of X₁₁₀

• X₃₀ = Lucas isogonal conjugate of X₁₁₀

• X₃₀ = homothetic center of X₂₀-altimedial and X₁₄₀-anti-altimedial triangles

• X₃₀ = X₁₁₅₄-of-excentral-triangle

• X₃₀ = homothetic center of Ehrmann vertex-triangle and Trinh triangle

• X₃₀ = homothetic center of Ehrmann side-triangle and dual of orthic triangle

• X₃₀ = homothetic center of Ehrmann mid-triangle and medial triangle

• X₃₀ = excentral-to-ABC functional image of X₅₁₇

• X₃₀ = 1st-Brocard-isogonal conjugate of X₁₈₃₃₂

• X₃₀ = polar conjugate of X₁₆₀₈₀

• X₃₀ = X₆₃-isoconjugate of X₈₇₄₉

Notes

As a point on the Euler line, X₃₀ has Shinagawa coefficients \((1, -3)\).

Let Ref_aRef_bRef_c be the reflection triangle. Let A″ be the cevapoint of Ref_b and Ref_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ are parallel to the Euler line, and therefore concur in X₃₀. (Randy Hutson, December 10, 2016)