X2999

DANNEELS-BEVAN HOMOTHETIC CENTER

Let \(R_{A}\) be the radical axis of the Bevan circle and the A-excircle. Define \(R_{B}\) and \(R_{C}\) cyclically. The three radical axes form a triangle homothetic to triangle ABC, and X2999 is the center of homothety. (Eric Danneels, Hyacinthos #10926, Dec. 5, 2004.
< 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) { 1, 2 }: Nagel Line { 6, 57 }: Central Line L(6, 57) { 31, 165 }: Central Line L(31, 165) { 58, 937 }: Central Line L(58, 937) { 46, 1203 }: Central Line L(46, 1203) { 3, 1453 }: Central Line L(3, 1453) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2297 = -ISOCONJUGATE OF X(1191) (0.5417, 0.2080) X_1 = INCENTER (0.5714, 0.2333) X_2 = CENTROID (0.5374, 0.2000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_937 = ISOGONAL CONJUGATE OF X(936) (0.6143, 0.2920) X_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_1203 = 3rd SARAGOSSA POINT OF X(58) (0.6034, 0.2712) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1453 = -BETH CONJUGATE OF X(452) (0.6101, 0.2792) X_2999 = DANNEELS-BEVAN HOMOTHETIC CENTER (0.5973, 0.2586) A B C X2297 X1 X2 X6 X57 X31 X165 X58 X937 X46 X1203 X3 X1453 X2999 (a, b, c) = (5, 6, 7) | AB = 1

X2999 (DANNEELS-BEVAN HOMOTHETIC CENTER) 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 \(f(a,b,c) = (a + b + c)^{2} - 4bc\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(1.105547730017\)
▸ Alternate Coordinates1 form

  • Trilinears: \(bc - s^{2} : ca - s^{2} : ab - s^{2}\)

▸ Incident Central Lines19 lines

{1, 2}, {3, 1453}, {6, 57}, {31, 165}, {46, 1203}, {58, 937}, {63, 1743}, {73, 1467}, {244, 1282}, {278, 2331}, {440, 1040}, {673, 2258}, {748, 968}, {940, 1449}, {990, 1750}, {1191, 1697}, {1214, 2257}, {1376, 1386}, {1394, 1466}

▸ Functional Relations3 relations

Notes

Let BA be the inverse-in-the-A-excircle of the B-excenter and CA the inverse-in-the-A-excircle of the C-excenter. Let LA be the line BACA, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The triangle A'B'C' is homothetic to triangle ABC, and the center of homothety is X2999. (Randy Hutson, 9/23/2011)