HYACINTHOS 28595

[Seiichi Kirikami]:

 

Dear friends,

Let ABC be a triangle and H its orthocenter. Let P be a triangle center of ABC.

Denote:

Pa, Pb, Pc = P of HBC, HCA, HAB respectively.

Attached are triangle centers P = X(m) in ETC such that HP, APa, BPb and CPc concur in a point Q = X(n). (m =< 3050.)

For an example of P = X(2), HP, APa, BPb and CPc concur in Q = X(5).

In case of n= 5, 51, 52, 53, 128, 129, 130, 134, 137, 138, 139, 143, 1263, P, Pa, Pb and Pc coincide with each other.

Problem 1 : what is an algebraic or synthetic condition that the lines HP, APa, BPb and CPc concur?

The following are not in ETC.

(6-9-13) search numbers:

Y1 = (3.7557081332.., -0.4976799279.., 2.2518083705..)

Y2 = (1.6416351771.., -0.5318879892.., 3.2512168543..)

Y3 = (2.5870847056.., -0.845294059.., 3.0318289663..)

Y4 = (0.4349864256..,-0.2739394486.., 3.6295519037..)

Y5 = (1.1475819483.., -0.4722319892.., 3.439411137..)

Y6 = (-4.8216718324.., -5.4215123459..,9.619406185..)

Y7 = (-0.4326943465..,0.329501169.., 3.612253371..)

Y8 = (0.0251493854.., 0.0286556003.., 3.6092185807..)

Y9 = (-0.5769841506.., 1.7026420687.., 2.7282126577..)

Y10 = (-0.5446463345.., 2.3280496715.., 2.2803130176..)

Y11 = (-1.0485432877.., 0.2818836923.., 3.9294572892..)

Y12 = (-4.5549921188.., -6.223634038.., 10.0516382553..)


Problem 2 : If P is any of Y1 – Y12, do P, Pa, Pb and Pc coincide with each other?

Best regards, Seiichi.


[Angel Montesdeoca]:
 

Dear Seiichi

*** P = X(157) , Q = Y1

Y1 = X(4)X(157)∩X(1899)X(2165)

= (a^2 (b^2+c^2)-(b^2-c^2)^2)/(a^8-2 a^6 (b^2+c^2)+2 a^4 (b^2+c^2)^2-2 a^2 (b^6+c^6)+(b^2-c^2)^2 (b^4+c^4)) : :

= lies on these lines: {4,157}, {1899, 2165}


*** P = X(323),  Q = Y2

Y2 = X(4)X(323)∩X(30)X(2980)

= ((b^2-c^2)^2-a^2 (b^2+c^2))/(a^6-3 a^4 (b^2+c^2)+a^2 (3 b^4-4 b^2 c^2+3 c^4)-(b^2-c^2)^2 (b^2+c^2))  : :

 
= lies on these lines: {4,323}, {30,2980}, {53,1154}, {97,1141}, {525,10412}, {2165,2549}, {8800,11591}, {13450,14918}
 

*** P= X(325), Q = Y3  

Y3 = X(4)X(325)∩X(562)X(6751)

= (a^2-b^2-c^2)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^8-6 a^6 b^2+4 a^4 b^4-2 a^2 b^6+b^8-6 a^6 c^2+8 a^4 b^2 c^2+2 a^2 b^4 c^2-4 b^6 c^2+4 a^4 c^4+2 a^2 b^2 c^4+6 b^4 c^4-2 a^2 c^6-4 b^2 c^6+c^8)  : :  

= lies on these lines: {4,325}, {562,6751}

 

*** P = X(373) ,  Q = Y4 

Y4 = MIDPOINT OF X(3855) AND X(15024)

 = a^2 (a^4+b^4-10 b^2 c^2+c^4-2 a^2 (b^2+c^2)) (a^2 (b^2+c^2)-(b^2-c^2)^2) : : 

= lies on these lines: {2,13348}, {4,373}, {5,51}, {20,6688}, {185,3091}, {381,11381}, {389,3545}, {511,5056}, {547,10625}, {632,12046}, {1216,5079}, {1495,7529}, {1656,5447}, {1995,13367}, {3060,15022}, {3066,11479}, {3090,3917}, {3146,13570}, {3543,17704}, {3544,3567}, {3819,7486}, {3832,9729}, {3839,15028}, {3843,5892}, {3850,9730}, {3851,5462}, {3854,10574}, {3855,6000}, {3857,12006}, {3858,10575}, {3859,13491}, {3861,14855}, {5020,11424}, {5055,5446}, {5066,12162}, {5067,15644}, {5068,5640}, {5071,9781}, {5072,13754}, {5198,17825}, {5642,15465}, {5651,10982}, {5876,11737}, {5946,12811}, {6101,10109}, {6467,14561}, {7398,19467}, {7528,11572}, {7544,13851}, {8227,16980}, {10219,10303}, {10263,12812}, {10545,14118}, {10594,22352}, {11414,22112}, {11444,21849}, {13474,15045}, {14269,14641}, {14831,19709}, {14892,16881}, {15004,17814}, {15056,16625}, {18369,18475}

= midpoint of X(3855) and X(15024)

 

*** P = X(294), Q = Y5

Y5 = X(4)X(394)∩X(53)X(5562)

= ((b^2-c^2)^2-a^2 (b^2+c^2))/(a^6+3 a^2 (b^2-c^2)^2-3 a^4 (b^2+c^2)-(b^2-c^2)^2 (b^2+c^2)) :  : 

= lies on these lines: {4,394}, {53,5562}, {343,13450}, {2165,5254}, {5891,8800}


*** P= X(1117), Q = Y6

Y6 = X(4)X(1117)∩X(137)X(24772)

=  a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2+a^2 b^2 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6)^2 (a^6-3 a^4 b^2+3 a^2 b^4-b^6-a^4 c^2+a^2 b^2 c^2+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6)^2 : : 

= lies on these lines: {4,1117}, {137,24772}, {1154,1263}, {1291,5899}, {13582,19552}


*** P = X(1970), Q = Y7

Y7 = X(4)X(1970)∩X(5)X(53)

=  (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4 b^4-2 a^2 b^6+b^8-a^4 b^2 c^2+2 a^2 b^4 c^2-b^6 c^2+a^4 c^4+2 a^2 b^2 c^4-2 a^2 c^6-b^2 c^6+c^8) : :

= lies on these lines: {4,1970}, {5,53}, {52,129}


*** P = X(1987), Q= Y8

Y8 = X(4)X(1987)∩X(5)X(53)

 =  (-a^2+b^2-c^2) (a^2+b^2-c^2) (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^8+2 a^6 b^2-a^4 b^4+2 a^6 c^2-3 a^4 b^2 c^2+b^6 c^2-a^4 c^4-2 b^4 c^4+b^2 c^6) : :

= lies on these lines: {4,1987}, {5,53}, {6,1093}, {217,13450}, {436,1970}, {458,15466}, {6748,10110}


*** P = X(1988 ), Q = Y9

Y9 = (name pending)

 =  (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^6 b^2-2 a^4 b^4+a^2 b^6-a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+2 a^4 c^4-a^2 b^2 c^4+2 b^4 c^4-a^2 c^6-b^2 c^6)^2 (a^6 b^2-2 a^4 b^4+a^2 b^6-a^6 c^2-a^4 b^2 c^2+a^2 b^4 c^2+b^6 c^2+2 a^4 c^4-a^2 b^2 c^4-2 b^4 c^4-a^2 c^6+b^2 c^6)^2,(a^2-b^2-c^2) (a^2+b^2-c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) (a^6 b^2-2 a^4 b^4+a^2 b^6-a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+2 a^4 c^4-a^2 b^2 c^4+2 b^4 c^4-a^2 c^6-b^2 c^6)^2 (a^6 b^2-2 a^4 b^4+a^2 b^6+a^6 c^2+a^4 b^2 c^2-a^2 b^4 c^2-b^6 c^2-2 a^4 c^4-a^2 b^2 c^4+2 b^4 c^4+a^2 c^6-b^2 c^6)^2 : : 

= lies on this line: {4,1988}   


*** P = X(1994 ), Q = Y10

Y10 = X(4)X(1994)∩X(5)X(15869)

 =  (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^6+3 a^4 b^2-3 a^2 b^4+b^6+a^4 c^2-3 b^4 c^2+a^2 c^4+3 b^2 c^4-c^6) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6) : : 

= lies on these lines: {4,1994}, {5,15869}, {26,2165}, {30,22261}, {53,143}, {343,565}, {1141,8883}, {1154,8800}, {3459,7488}, {5576,16837}, {7540,11816}, {8905,25150}, {10279,10412}, {13450,14129}


*** P = X(2351), Q = Y11

Y11 = X(4)X(2351)∩X(5)X(51)

=  (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) (a^4+b^4-2 b^2 c^2+c^4) : : 

= lies on these lines: {4,2351}, {5,51}, {129,134}, {132,135}, {138,139}, {230,427}, {1899,2450}, {5133,9753}, {11245,23333}


*** P = X( 2413), Q = Y12 

Y12 = X(4)X(24139)∩X(143)X(1510)

 = (b-c) (b+c) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (-a^16+4 a^14 b^2-6 a^12 b^4+4 a^10 b^6-a^8 b^8+4 a^14 c^2-10 a^12 b^2 c^2+8 a^10 b^4 c^2-2 a^8 b^6 c^2+a^4 b^10 c^2-2 a^2 b^12 c^2+b^14 c^2-6 a^12 c^4+8 a^10 b^2 c^4-3 a^8 b^4 c^4+a^4 b^8 c^4+6 a^2 b^10 c^4-6 b^12 c^4+4 a^10 c^6-2 a^8 b^2 c^6-4 a^4 b^6 c^6-4 a^2 b^8 c^6+15 b^10 c^6-a^8 c^8+a^4 b^4 c^8-4 a^2 b^6 c^8-20 b^8 c^8+a^4 b^2 c^10+6 a^2 b^4 c^10+15 b^6 c^10-2 a^2 b^2 c^12-6 b^4 c^12+b^2 c^14) : : 

 = lies on these lines: {4,24139}, {143,1510}

 Best regards,
 Angel Montesdeoca