X984

MIDPOINT OF X8 AND X192

X984 is the midpoint of X8 and X192.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 8, 192 }: Central Line L(8, 192) { 1, 6 }: Incenter–Symmedian Line { 2, 38 }: Central Line L(2, 38) { 10, 75 }: Central Line L(10, 75) { 63, 171 }: Central Line L(63, 171) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_8 = NAGEL POINT (0.4694, 0.1333) X_192 = -CEVA CONJUGATE OF X(2) (CONGRUENT PARALLELIANS POINT) (0.6147, 0.2635) X_1 = INCENTER (0.5714, 0.2333) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_2 = CENTROID (0.5374, 0.2000) X_38 = CROSSPOINT OF X(1) AND X(75) (0.5443, 0.1976) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_75 = ISOTOMIC CONJUGATE OF INCENTER (0.4988, 0.1682) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.5819, 0.2525) X_984 = MIDPOINT OF X(8) AND X(192) (0.5421, 0.1984) A B C X8 X192 X1 X6 X2 X38 X10 X75 X63 X171 X984 (a, b, c) = (5, 6, 7) | AB = 1

X984 (MIDPOINT OF X8 AND X192) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(b^{2} + bc + c^{2} : :\)
Search Value (6, 9, 13): \(2.491338244862\)
▸ Incident Central Lines15 lines

{1, 6}, {2, 38}, {8, 192}, {10, 75}, {21, 976}, {43, 210}, {55, 846}, {63, 171}, {100, 753}, {101, 761}, {201, 388}, {240, 281}, {386, 872}, {519, 751}, {936, 988}

▸ Functional Relations8 relations

Notes

Let A15B15C15 and A16B16C16 be Gemini triangles 15 and 16, resp. Let LA be the tangent at A to conic {A,B15,C15,B16,C16}}, and define LB, LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X984. (Randy Hutson, January 15, 2019)