HYACINTHOS 25881

[Tran Quang Hung]:

 

Let ABC be a triangle. The mixtilinear incircle touches circumcircle (O) at A',B',C'. A circle other than (O) touches three mixtilinear incircles internally at A'',B'',C''.

Let B'C'' cuts C'B'' at A*. Similarly, we have B*,C*. Then AA*,BB*,CC* are concurrent on OI line of ABC.

Romantics of Geometry #712

[Peter Moses]:

 

Hi Antreas,

 

A’ = {-a (a+b-c) (a-b+c),2 b^2 (a+b-c),2 c^2 (a-b+c)}.

A’’ = {-a (a^2+2 a b-3 b^2+2 a c+6 b c-3 c^2),2 b^2 (-a+b-c),-2 (a+b-c) c^2}.

A* = {-8 a^2 (a^2-b^2+b c-c^2),-b (-a+b-c) (-3 a^2+2 a b+5 b^2+6 a c+2 b c-3 c^2),-(a+b-c) c (3 a^2-6 a b+3 b^2-2 a c-2 b c-5 c^2)}.

 

Perspector ABC & A*B*C* =

a (a-b-c) (5 a^2+2 a b-3 b^2+2 a c+6 b c-3 c^2):: 

on lines {{1,3},{2,5727},{9,2320},{78, 3897},{200,4711},{226,5731},{ 390,6173},{392,10391},{495, 3655},{497,551},{498,5881},{ 515,5219},{519,5218},{944, 6956},{946,4305},{950,3616},{ 991,1457},{993,3929},{997, 7308},{1125,3486},{1149,2293}, {1317,9952},{1419,1455},{1479, 9624},{1698,5326},{1706,4855}, {1837,3624},{2136,4861},{2268, 3247},{2975,11523},{3085,5882} ,{3158,3872},{3241,5281},{ 3306,4881},{3475,4315},{3485, 4297},{3487,4311},{3522,4323}, {3523,4848},{3524,11041},{ 3577,6905},{3586,5886},{3622, 4313},{3636,4314},{3679,5432}, {3680,3871},{3689,4915},{4189, 11682},{4293,4654},{4304,5603} ,{4342,10385},{4512,5289},{ 4845,11712},{4853,8168},{4870, 12943},{5270,10953},{5284, 5436},{5332,9575},{5424,7284}, {5440,9623},{5587,6859},{5691, 11375},{5703,10106},{5901, 9614},{6284,11522},{6738,7288} ,{6867,8227},{6906,7971},{ 7675,10384},{8167,8583},{9613, 11374},{10527,12625},{10543, 11376},{10595,10624}}.

3 X[5219] - 2 X[10590].

(3/2 + R/r) X[1] + X[3].

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,3340),(1,35,7982),(1,36, 11529),(1,55,7962),(1,56, 11518),(1,165,2099),(1,1385, 1420),(1,2646,3601),(1,3576, 57),(1,3601,1697),(1,3612,40), (1,7987,65),(1,7991,11011),(1, 9819,5048),(3,3340,5128),(36, 11529,57),(55,5048,9819),(55, 7962,1697),(1125,3486,9581),( 3485,4297,9579),(3576,11529, 36),(3601,7962,55),(3622,4313, 12053),(4304,5603,9580),(5048, 9819,7962),(5217,11011,7991).

Searches: {2.4296267504355177110,2. 3007829076093917718,0. 92644858413071508678}.

 

A*B*C* is also perspective to A’’B’’C’’ and the 1st and 3rd (=A’B’C’) mixtilinear trangles ... also all on the OI line.

 

Best regards,

Peter Moses.