HYACINTHOS 27056

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the cevian triangle of O.

Denote:

Ab, Ac = the orthogonal projections of A' on BB', CC', resp.
A2, A3 = the orthogonal projections of Ab, Ac on BC, resp.

Bc, Ba = the orthogonal projections of B' on CC', AA', resp.
B3, B1 = the orthogonal projections of Bc, Ba on CA, resp.

 

Ca, Cb = the orthogonal projections of C' on AA', BB', resp.
C1, C2 = the orthogonal projections of Ca, Cb on AB, resp.

M1, M2, M3 = the midpoints of B1C1, C2A2, A3B3, resp.

 

ABC, M1M2M3 are perspective.


[Peter Moses]:

Hi Antreas,

X(343).

ABC & M1M2M3 are perspective for the cevian triangle, A'B'C', of a point on K004.

 

[ie  The locus of P such that the cevian triangle A'B'C' of P and M1M2M3 are perspective is the Darboux cubic K004.

P = I. Hyacinthos 27037 

P = H. Hyacinthos 27039  ]

 

For example, A'B'C' as the cevian of X(20) -> 

a^2 (a^2-b^2-c^2) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4):: = a^2 SA (S^2 - 2 SB SC)::
on lines {{3,6},{5,3087},{20,1249},{25,10313},{30,393},{48,222},{53,382},{69,441},{97,5422},{109,3197},{112,1033},{159,1576},{219,255},{221,1630},{232,9909},{233,5070},{248,6391},{381,6748},{394,1073},{401,9308},{418,11402},{524,6389},{607,1950},{608,1951},{610,1394},{852,6090},{1012,1172},{1060,3553},{1062,3554},{1368,7735},{1396,11347},{1415,1604},{1498,3348},{1589,7585},{1590,7586},{1598,14152},{1656,6749},{1657,1990},{1660,1661},{1971,8780},{2072,9722},{2189,13730},{2289,7078},{2968,5839},{3079,3344},{3129,11409},{3130,11408},{3148,12167},{3155,5411},{3156,5410},{3163,15681},{3167,3289},{3194,9122},{3522,5702},{3964,4558},{4644,6356},{5020,10311},{5286,12362},{5304,7386},{5305,6643},{6413,6415},{6414,6416},{6641,9777},{6642,8882},{6676,7736},{7387,8745},{7517,14577},{8743,11414},{8744,12082},{8746,12083},{8778,11413},{9475,12220},{9714,14576},{15484,15760}}.
isogonal conjugate of X(459).
X(2327)-beth conjugate of X(268).
X(1790)-gimel conjugate of X(5120).
X(15077)-complementary conjugate of X(2887).
X(i)-Ceva conjugate of X(j) for these (i,j): {{20, 154}, {394, 3}, {2327, 48}}.
X(154)-cross conjugate of X(3).
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10317, 1384), (3, 15851, 216), (6, 50, 1609), (6, 216, 15851), (6, 571, 8573), (6, 577, 3), (6, 3053, 800), (371, 372, 9786), (394, 6617, 1073), (571, 8573, 1384), (577, 3284, 6), (1151, 1152, 1620), (3311, 3312, 11432).
X(i)-isoconjugate of X(j) for these (i,j): {{1, 459}, {4, 2184}, {19, 253}, {63, 6526}, {64, 92}, {158, 1073}, {264, 2155}, {281, 8809}, {1301, 1577}, {1880, 5931}, {2190, 13157}, {6520, 15394}, {6521, 14379}}.
crosspoint of X(i) and X(j) for these (i,j): {{2, 15077}, {250, 4558}, {1262, 1331}}.
crossdifference of every pair of points on line X(523) X(10151).
crosssum of X(i) and X(j) for these (i,j): {{6, 3515}, {125, 2501}, {393, 6526}, {523, 1562}, {1146, 7649}, {1990, 13202}}.
barycentric product X(i)X(j) for these {i,j}: {{3, 20}, {63, 610}, {69, 154}, {77, 7070}, {78, 1394}, {110, 8057}, {122, 250}, {184, 14615}, {204, 326}, {249, 1562}, {255, 1895}, {283, 5930}, {394, 1249}, {577, 15466}, {1092, 14249}, {1444, 3198}, {1790, 8804}, {1813, 14331}, {2060, 3348}, {3079, 15394}, {3172, 3926}, {3213, 3719}, {3344, 6617}, {3964, 6525}, {4558, 6587}, {7156, 7183}, {11064, 15291}}.
barycentric quotient X(i)/X(j) for these {i,j}: {{3, 253}, {6, 459}, {20, 264}, {25, 6526}, {48, 2184}, {122, 339}, {154, 4}, {184, 64}, {204, 158}, {216, 13157}, {283, 5931}, {418, 8798}, {577, 1073}, {603, 8809}, {610, 92}, {1092, 15394}, {1249, 2052}, {1394, 273}, {1562, 338}, {1576, 1301}, {3079, 14249}, {3172, 393}, {6525, 1093}, {6587, 14618}, {7070, 318}, {8057, 850}, {9247, 2155}, {14585, 14642}}.

Best regards,
Peter Moses.