X₁₄₀

MIDPOINT OF X₃ AND X₅

X₁₄₀ is the midpoint of X₃ and X₅.

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

X₁₄₀ (MIDPOINT OF X₃ AND X₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 5} (Euler line)

• Related Centers: X₃ (CIRCUMCENTER), X₅ (NINE-POINT CENTER)

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

Barycentrics: \(b \cos(C - A) + c \cos(B - A) : :\)

Search Value (6, 9, 13): \(3.667617317916\)

▸ Alternate Coordinates7 forms

• Trilinears: \(\cos A + 2 \sin B \sin C : \cos B + 2 \sin C \sin A : \cos C + 2 \sin A \sin B\)

• Trilinears: \(3 \cos A + 2 \cos B \cos C : 3 \cos B + 2 \cos C \cos A : 3 \cos C + 2 \cos A \cos B\)

• Trilinears: \(2 \sec A + 3 \sec B \sec C : 2 \sec B + 3 \sec C \sec A : 2 \sec C + 3 \sec A \sec B\)

• Trilinears: \(bc[b \cos(C - A) + c \cos(B - A)] : :\)

• Barycentrics: \(3 - \cot B \cot C : :\)

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

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

▸ Linear Combinations1 combination

• X₁₄₀ = \(3\) X₂ + X₃ = \(3\) X₂ - X₅ = \(3\) X₃ + X₄ = X₄ - \(3\) X₅

▸ Incident Central Lines399 lines

{1, 5432}, {2, 3}, {6, 5418}, {8, 1483}, {9, 5843}, {10, 214}, {11, 35}, {12, 36}, {13, 5237}, {14, 5238}, {15, 18}, {16, 17}, {32, 3815}, {39, 230}, {40, 3624}, {46, 11375}, {49, 5012}, {51, 10263}, {52, 3917}, {53, 10979}, {54, 252}, {55, 496}, {56, 495}, {57, 6147}, {61, 395}, {62, 396}, {69, 1353}, {72, 10202}, {76, 6390}, {79, 5131}, {83, 2080}, {95, 340}, {98, 7832}, {100, 1484}, {104, 5260}, {110, 10264}, {113, 10990}, {114, 6292}, {115, 10992}, {119, 5251}, {125, 128}, {141, 182}, {142, 5762}, {143, 511}, {156, 9306}, {165, 8227}, {183, 3933}, {184, 13336}, {185, 5876}, {187, 1506}, {195, 323}, {216, 1990}, {231, 570}, {233, 6748}, {236, 8129}, {262, 7846}, {298, 628}, {299, 627}, {302, 633}, {303, 634}, {325, 1078}, {343, 569}, {355, 1698}, {371, 615}, {372, 590}, {385, 13571}, {389, 1154}, {392, 11729}, {394, 12161}, {484, 5443}, {485, 1152}, {486, 1151}, {497, 10386}, {515, 3634}, {516, 9955}, {517, 1125}, {518, 13373}, {523, 1116}, {524, 575}, {539, 11264}, {542, 6698}, {551, 10222}, {567, 3580}, {568, 7998}, {572, 1213}, {574, 3054}, {576, 597}, {577, 6749}, {578, 13567}, {601, 748}, {602, 750}, {618, 630}, {619, 629}, {620, 2782}, {623, 13350}, {624, 13349}, {625, 7830}, {626, 13335}, {671, 10185}, {758, 5885}, {908, 3916}, {912, 5044}, {930, 1263}, {936, 5791}, {942, 3911}, {944, 5790}, {946, 3579}, {956, 5552}, {958, 3820}, {970, 6703}, {971, 6666}, {993, 1329}, {999, 3085}, {1001, 10200}, {1007, 3785}, {1040, 8144}, {1056, 5265}, {1058, 5281}, {1071, 12691}, {1141, 11016}, {1145, 4861}, {1155, 12047}, {1173, 12834}, {1210, 12433}, {1319, 10039}, {1351, 3618}, {1352, 3763}, {1376, 10267}, {1387, 3057}, {1388, 12647}, {1478, 5204}, {1479, 5217}, {1482, 3616}, {1493, 13366}, {1503, 5092}, {1587, 6398}, {1588, 6221}, {1621, 11849}, {1697, 11373}, {1737, 2646}, {1768, 3652}, {1834, 4256}, {1837, 3612}, {1853, 9833}, {2077, 5259}, {2095, 5761}, {2548, 3053}, {2777, 5893}, {2794, 6721}, {2800, 13145}, {2808, 6710}, {2818, 6711}, {2831, 11259}, {2883, 3357}, {2888, 7666}, {2896, 7925}, {2979, 3567}, {3019, 13329}, {3068, 3312}, {3069, 3311}, {3070, 6396}, {3071, 6200}, {3086, 3295}, {3095, 7786}, {3096, 7940}, {3098, 5480}, {3167, 11411}, {3216, 5396}, {3303, 10072}, {3304, 10056}, {3316, 3590}, {3317, 3591}, {3336, 3649}, {3337, 5557}, {3419, 4855}, {3487, 5435}, {3488, 5704}, {3532, 4846}, {3581, 5888}, {3582, 3746}, {3583, 7173}, {3584, 5298}, {3585, 3614}, {3600, 8164}, {3601, 5722}, {3617, 7967}, {3619, 6776}, {3620, 11898}, {3622, 10247}, {3626, 13607}, {3630, 7916}, {3631, 5965}, {3653, 3679}, {3654, 7982}, {3655, 5881}, {3656, 7991}, {3678, 12005}, {3740, 12675}, {3767, 5013}, {3793, 7762}, {3813, 8715}, {3814, 5267}, {3816, 5248}, {3822, 5841}, {3824, 12436}, {3825, 5840}, {3826, 6796}, {3841, 5842}, {3898, 10284}, {3925, 10902}, {3927, 5744}, {4045, 7886}, {4255, 5292}, {4292, 5122}, {4293, 9654}, {4294, 9669}, {4297, 10175}, {4299, 10895}, {4302, 10896}, {4309, 11238}, {4317, 11237}, {4413, 11499}, {4423, 10310}, {5007, 9300}, {5010, 6284}, {5023, 7737}, {5024, 5286}, {5045, 13405}, {5080, 5303}, {5086, 10609}, {5097, 6329}, {5119, 11376}, {5126, 10106}, {5157, 13562}, {5171, 7808}, {5188, 7889}, {5206, 5475}, {5306, 7772}, {5309, 9607}, {5318, 10646}, {5321, 10645}, {5339, 11480}, {5340, 11481}, {5398, 5718}, {5403, 8160}, {5404, 8161}, {5414, 9661}, {5437, 5709}, {5438, 5705}, {5440, 6734}, {5446, 5943}, {5449, 12038}, {5486, 8548}, {5489, 5664}, {5534, 8580}, {5550, 5603}, {5562, 5650}, {5569, 7775}, {5587, 7987}, {5590, 5874}, {5591, 5875}, {5609, 5642}, {5640, 11465}, {5646, 5654}, {5651, 10539}, {5656, 13093}, {5658, 12684}, {5663, 5907}, {5687, 10527}, {5694, 5884}, {5720, 8726}, {5731, 5818}, {5743, 13323}, {5777, 11227}, {5878, 10606}, {5883, 11281}, {5889, 7999}, {5890, 11444}, {5894, 11204}, {5944, 12134}, {6000, 6696}, {6033, 7944}, {6055, 9167}, {6130, 8552}, {6153, 13433}, {6194, 7875}, {6199, 7582}, {6247, 6759}, {6248, 7820}, {6287, 9751}, {6321, 7847}, {6368, 10213}, {6395, 7581}, {6409, 6561}, {6410, 6560}, {6417, 7586}, {6418, 7585}, {6425, 9680}, {6449, 6459}, {6450, 6460}, {6455, 9541}, {6502, 9646}, {6592, 8902}, {6669, 6673}, {6670, 6674}, {6688, 10110}, {6722, 7861}, {6723, 11801}, {6746, 12363}, {7028, 8130}, {7280, 7354}, {7308, 7330}, {7603, 7747}, {7610, 12040}, {7616, 8859}, {7622, 7781}, {7697, 7835}, {7709, 7891}, {7735, 9605}, {7743, 10624}, {7750, 7752}, {7751, 13468}, {7756, 8589}, {7758, 8667}, {7761, 7862}, {7778, 7800}, {7801, 11168}, {7806, 12251}, {7810, 7821}, {7811, 7814}, {7831, 7899}, {7834, 9737}, {7844, 9734}, {7854, 7888}, {7863, 9466}, {7868, 9744}, {7881, 9755}, {7904, 7912}, {7914, 9996}, {7956, 8167}, {8071, 10320}, {8125, 8128}, {8126, 8127}, {8141, 10319}, {8148, 10595}, {8151, 10190}, {8666, 12607}, {8721, 9756}, {8722, 10358}, {9301, 10357}, {9655, 10590}, {9668, 10591}, {9704, 11003}, {9707, 11457}, {9729, 9820}, {9781, 11451}, {9826, 13416}, {10006, 11247}, {10163, 12506}, {10171, 12512}, {10189, 10280}, {10198, 11249}, {10225, 11813}, {10277, 13582}, {10517, 11916}, {10518, 11917}, {10541, 11179}, {10572, 12019}, {10574, 11459}, {10584, 11928}, {10585, 11929}, {10586, 12000}, {10587, 12001}, {10628, 11561}, {10915, 11260}, {10950, 11545}, {11015, 12690}, {11017, 13474}, {11176, 11615}, {11219, 12738}, {11246, 11544}, {11255, 11511}, {11265, 11513}, {11266, 11514}, {11267, 11515}, {11268, 11516}, {11426, 11433}, {11427, 11432}, {11430, 12241}, {11438, 12233}, {11485, 11489}, {11486, 11488}, {11703, 11792}, {11808, 13365}, {12162, 13491}, {12325, 13432}, {12358, 13148}, {13142, 13352}, {13470, 13565}

▸ Functional Relations34 relations

• X₁₄₀ = midpoint of X_i and X_j for these (i,j): (3, 5), (141, 182), (389, 1216), (2883, 3357)

• X₁₄₀ = reflection of X_i in X_j for these (i,j): (546, 5), (547, 2), (548, 3)

• X₁₄₀ = isogonal conjugate of X₁₁₇₃

• X₁₄₀ = complement of X₅

• X₁₄₀ = anticomplement of X₃₆₂₈

• X₁₄₀ = complementary conjugate of X₁₂₀₉

• X₁₄₀ = polar-circle-inverse of X₃₇₉₄₃

• X₁₄₀ = X₂-Ceva conjugate of X₂₃₃

• X₁₄₀ = crosspoint of X_i and X_j for these (i,j): (2, 95), (17, 18)

• X₁₄₀ = crosssum of X_i and X_j for these (i,j): (6, 51), (61, 62)

• X₁₄₀ = crosspoint of the two Napoleon points, X₁₇ and X₁₈

• X₁₄₀ = inverse-in-orthocentroidal-circle of X₁₆₅₆

• X₁₄₀ = X₅-of-medial triangle

• X₁₄₀ = centroid of the quadrangle ABCX₃

• X₁₄₀ = perspector of circumconic centered at X₂₃₃

• X₁₄₀ = center of circumconic that is locus of trilinear poles of lines passing through X₂₃₃

• X₁₄₀ = intersection of tangents to Evans conic at X₃₀₇₀ and X₃₀₇₁

• X₁₄₀ = centroid of X₂X₃X₁₁₅X₂₄₈₂

• X₁₄₀ = pole of Brocard axis wrt conic {X₅,X₁₃,X₁₄,X₁₅,X₁₆}}

• X₁₄₀ = X₃ of polar triangle of complement of polar circle

• X₁₄₀ = inverse-in-complement-of-polar-circle of X₂₀₇₂

• X₁₄₀ = inverse-in-{circumcircle, nine-point circle}-inverter of X₅₁₈₉

• X₁₄₀ = center of inverse-in-{circumcircle, nine-point circle}-inverter of anticomplementary circle

• X₁₄₀ = centroid of the six circumcenters in the construction of the van Lamoen circle.

• X₁₄₀ = centroid of ABCX₃

• X₁₄₀ = Kosnita(X₃,X₂) point

• X₁₄₀ = center of circle that is locus of crosssums of antipodes on the 2nd Lemoine circle

• X₁₄₀ = {X₂,X₅}-harmonic conjugate of X₃₆₂₈

• X₁₄₀ = {X₃,X₄}-harmonic conjugate of X₅₅₀

• X₁₄₀ = {X₄,X₅}-harmonic conjugate of X₃₈₅₀

• X₁₄₀ = homothetic center of X₄-altimedial and X₁₄₀-anti-altimedial triangles

• X₁₄₀ = X₃₅₇₉-of-orthic-triangle if ABC is acute

• X₁₄₀ = pole of van Aubel line wrt conic {X₂,X₁₅,X₁₆,X₁₇,X₁₈}}

• X₁₄₀ = homothetic center of McCay and Moses-Steiner osculatory triangles

Notes

As a point on the Euler line, X₁₄₀ has Shinagawa coefficients \((3, -1)\).

Let A' be the midpoint between A and X₃, and define B' and C' cyclically; the triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X₁₄₀. Let A″ be the centroid of the triangle BCX₃, and define B″ and C″ cyclically; then A″B″C″ is homothetic to ABC, and the center of homothety is X₁₄₀. Also, X₁₄₀ is the center of the conic consisting of the centers of all the conics which pass through A, B, C, and X₃. (Randy Hutson, 9/23/2011) This conic is also the locus of crosssums of the intersections of the circumcircle and lines through X₄. Furthmore, this conic is the bicevian conic of X₂ and X₃. (Randy Hutson, 9/14/2016)

Let O_a be the circle centered at A and passing through the A-vertex of the Euler triangle; define O_b and O_c cyclically. The radical center of O_a, O_b, O_c is X₁₄₀. (Randy Hutson, September 14, 2016)

Let O_a be the circle centered at A with radius 1/2*\(\sqrt{b^{2} + c^{2}}\), and define O_b and O_c cyclically. The radical center of O_a, O_b, O_c is X₁₄₀. (Randy Hutson, September 14, 2016)

Let P be a point on the circumcircle. The bicevian conic of X₂ and P is a rectangular hyperbola, H. Let X be the center of H. As P varies, X traces a circle centered at X₁₄₀. (Randy Hutson, November 2, 2017)

X₁₄₀ is the centroid of the six circumcenters in the construction of the van Lamoen circle. (Randy Hutson, October 15, 2018)

X₁₄₀ is the {X₂,X₃}-harmonic conjugate of X₅.