HYACINTHOS 28741

[Aris Pavlakis](*)

Let ABC be a triangle and A'B'C' the pedal triangle of I.

The reflections of IN in B'C', C'A', A'B' bound a triangle A*B*C* similar to ABC.

The circumcenter O* of A*B*C* lies on the NPC.

Point?.  

 

(*) Aris Pavlakis, Romantics of Geometry 2583

[Peter Moses]:

Hi Antreas,

X(3259); (2*a - b - c)*(b - c)^2*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3) : : 

 

= lies on the nine point circle and these lines: {2, 901}, {4, 953}, {11, 513}, {12, 13756}, {25, 10016}, {36, 855}, {56, 17101}, {114, 1281}, {115, 661}, {116, 3835}, {119, 517}, {120, 5087}, {121, 3814}, {133, 5146}, {149, 14513}, {153, 14511}, {226, 24201}, {244, 6615}, {1319, 1846}, {1566, 6544}, {1647, 5516}, {1878, 22835}, {2969, 20620}, {3120, 7336}, {3326, 10017}, {4370, 5513}, {4404, 24026}, {5954, 5993}, {5957, 11792}, {7951, 23153}, {8286, 8819}, {15614, 21252}, {17036, 20096}, {17605, 23152}, {24250, 25760}

= anticomplement of X(22102)
= complement of X(901)
= midpoint of X(i) and X(j) for these {i,j}: {4, 953}, {56, 17101}, {149, 14513}, {153, 14511}
= reflection of X(i) and X(j) for these {i,j}: {901, 22102}, {3937, 14115}, {6073, 119}, {6075, 11}
= reflection of X(3937) in the X(1X(3) line
= on circumcircle of Yff contact triangle.
= {X(2),X(901)}-harmonic conjugate of X(22102)
= polar circle inverse of X(1309)
= orthoptic circle of the Steiner inellipe inverse of X(2726)
= complement of the isogonal of X(900)
= X(i)-complementary conjugate of X(j) for these (i,j): {1, 900}, {2, 4928}, {6, 3960}, {31, 3310}, {42, 21894}, {44, 514}, {244, 1647}, {513, 519}, {514, 3834}, {519, 513}, {522, 5123}, {649, 16610}, {661, 3936}, {667, 8610}, {678, 6544}, {693, 21241}, {765, 6550}, {876, 25351}, {900, 10}, {902, 650}, {1019, 4395}, {1023, 4422}, {1319, 522}, {1404, 905}, {1635, 2}, {1639, 3452}, {1647, 11}, {1877, 521}, {1960, 37}, {2087, 1086}, {2161, 21198}, {2251, 6586}, {2325, 20317}, {2429, 25097}, {3251, 4370}, {3264, 21260}, {3285, 14838}, {3669, 17067}, {3689, 4521}, {3762, 141}, {3911, 4885}, {3943, 4129}, {4120, 1211}, {4358, 3835}, {4448, 17793}, {4530, 26932}, {4730, 1213}, {4768, 1329}, {4893, 27751}, {4895, 9}, {5440, 20315}, {6544, 16594}, {14407, 16589}, {14408, 6376}, {14427, 6554}, {14429, 21530}, {14437, 13466}, {14584, 3738}, {16704, 4369}, {17780, 24003}, {21805, 661}, {22086, 1214}, {23344, 24036}, {23703, 3035}, {23757, 119}, {23838, 3036}, {24004, 27076}
= X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3310}, {4, 900}, {11, 1647}, {513, 6550}, {1145, 23757}
= X(i)-isoconjugate of X(j) for these (i,j): {104, 9268}, {765, 10428}, {909, 5376}
= crosspoint of X(i) and X(j) for these (i,j): {513, 517}, {900, 14584}, {1145, 23757}
= crosssum of X(100) and X(104)
= crossdifference of every pair of points on line {2423, 2427}
= barycentric product X(i) X(j) for these {i,j}: {514, 23757}, {900, 10015}, {908, 1647}, {1086, 1145}, {1769, 3762}, {1846, 26932}, {2087, 3262}, {2397, 6550}, {4120, 23788}, {4530, 22464}, {17205, 21942}
= barycentric quotient X(i) / X(j) for these {i,j}: {517, 5376}, {900, 13136}, {1015, 10428}, {1145, 1016}, {1769, 3257}, {2087, 104}, {2183, 9268}, {2397, 6635}, {2427, 6551}, {2804, 4582}, {3310, 901}, {6550, 2401}, {8661, 2423}, {10015, 4555}, {23757, 190}, {23788, 4615}

Best regards,
Peter.