X3052

INTERSECTION X31X_42∩X32X_220

X3052 = X56-Ceva conjugate of X6
< 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) { 6, 31 }: Central Line L(6, 31) { 31, 42 }: Central Line L(31, 42) { 44, 200 }: Central Line L(44, 200) { 32, 220 }: Central Line L(32, 220) { 3, 595 }: Central Line L(3, 595) { 45, 612 }: Central Line L(45, 612) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_44 = -LINE CONJUGATE OF X(1) (2.1429, 2.0996) X_200 = -CEVA CONJUGATE OF X(9) (0.4392, 0.1037) X_32 = 3rd POWER POINT (0.6400, 0.3332) X_220 = -CEVA CONJUGATE OF X(55) (0.4826, 0.1278) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_595 = 2nd HATZIPOLAKIS-YIU POINT (0.6456, 0.3307) X_45 = -BETH CONJUGATE OF X(1) (0.5418, 0.1981) X_612 = INTERSECTION OF LINES X(1)X(2) AND X(9)X(31) (0.5582, 0.2203) X_3052 = INTERSECTION X(31)X(42)∩X(32)X(220) (0.6825, 0.3888) A B C X6 X31 X42 X44 X200 X32 X220 X3 X595 X45 X612 X3052 (a, b, c) = (5, 6, 7) | AB = 1

X3052 (INTERSECTION X31X_42∩X32X_220) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(3a - b - c)\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-0.249098096131\)
▸ Incident Central Lines25 lines

{3, 595}, {6, 31}, {31, 42}, {32, 220}, {40, 1104}, {44, 200}, {45, 612}, {56, 1149}, {57, 1279}, {101, 1384}, {109, 1407}, {154, 2352}, {171, 1001}, {221, 1464}, {222, 2078}, {238, 1376}, {518, 1707}, {614, 1155}, {692, 1397}, {1260, 2220}, {1333, 2256}, {1460, 1486}, {1836, 3011}, {2176, 2223}, {2192, 2342}

▸ Functional Relations5 relations

Notes

Let TanaTanbTanc be the tangential triangle of ABC, and let L be the line through X1 parallel to BC. Let A″ = L∩TanbTanc, and define B″ and C″ cyclically. Let A* = TanbB″∩TancC″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X*(3445), and the lines TanaA*, TanbB*, TancC* concur in X3052. (Angel Montesdeoca, April 29, 2016)