X4000

X346com[T(-a, c)]

As described in the preamble to X3758, the notation T(-a, c) represents the central triangle whose A-vertex is the point \(-a : c : b\). This triangle and seven others are discussed just before X4357.
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 346 }: Central Line L(2, 346) { 6, 7 }: Central Line L(6, 7) { 2, 37 }: Central Line L(2, 37) { 19, 57 }: Central Line L(19, 57) { 1, 142 }: Central Line L(1, 142) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5374, 0.2000) X_346 = ISOTOMIC CONJUGATE OF X(279) (0.3948, 0.0827) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_7 = GERGONNE POINT (0.5903, 0.2769) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_19 = CLAWSON POINT (0.6246, 0.4244) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_1 = INCENTER (0.5714, 0.2333) X_142 = COMPLEMENT OF X(9) (0.5506, 0.2192) X_4000 = com[T(-a, c)] (0.6087, 0.2586) A B C Ma Mb Mc X2 X346 X6 X7 X37 X19 X57 X1 X142 X4000 (a, b, c) = (5, 6, 7) | AB = 1

X4000 (X346com[T(-a, c)]) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \(a^{2} + b^{2} + c^{2} - 2bc : :\)
Search Value (6, 9, 13): \(1.139393143412\)
▸ Alternate Coordinates2 forms

  • Barycentrics: \(\cot^{2}(B/2) + \cot^{2}(C/2) : :\)

  • Barycentrics: \(b c - S_{\omega} : :\)

▸ Incident Central Lines63 lines

{1, 142}, {2, 37}, {3, 1612}, {4, 990}, {6, 7}, {8, 141}, {9, 3008}, {10, 4353}, {19, 57}, {20, 1104}, {27, 2221}, {31, 3474}, {42, 3475}, {44, 144}, {48, 1429}, {69, 239}, {77, 3554}, {105, 1486}, {145, 3834}, {172, 4209}, {193, 320}, {226, 2999}, {241, 347}, {273, 393}, {279, 1418}, {319, 3620}, {329, 3782}, {348, 2275}, {386, 2140}, {387, 942}, {388, 4327}, {390, 1279}, {497, 614}, {499, 1733}, {518, 4310}, {527, 1743}, {594, 3763}, {894, 3618}, {910, 3598}, {938, 1834}, {962, 1191}, {1100, 3945}, {1107, 4352}, {1125, 4356}, {1193, 3485}, {1210, 1861}, {1266, 3729}, {1386, 4307}, {1423, 2183}, {1449, 3664}, {1453, 1890}, {1471, 4331}, {1473, 1851}, {1618, 2175}, {1714, 3670}, {1722, 2551}, {1760, 3218}, {3086, 4008}, {3212, 3959}, {3247, 4021}, {3486, 3924}, {3619, 3661}, {3875, 3912}

▸ Functional Relations5 relations

Notes

X4000 is the center of the inellipse that is the barycentric square of the Gergonne line. The Brianchon point (perspector) of the inellipse is X279. (Randy Hutson, October 15, 2018)