HYACINTHOS 29439

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle and P a point.

Denote:

A', B', C' = the midpoints of AP, BP, CP, resp.

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

(Oa), (Ob), (Oc) = the circles with diameters A'Na, B'Nb, C'Nc, resp.

Ra = the radical axis of (Ob), (Oc)
Rb = the radical axis of (Oc), (Oa)
Rc = the radical axis of (Oa), (Ob)

R1, R2, R3 = the reflections of of Ra, Rb, Rc in NbNc, NcNa, NaNb, resp.

1. R1, R2, R3 are concurrent.
Point of concurrence in terms of P?
Note: A question is: 
Which is the locus of P such that the point of concurrence lies on the Euler line?

2. For P = I:
(In this case Ra, Rb, Rc are the altitude lines of the pedal triangle of I)
The reflections of R1, R2, R3 in BC, CA, AB, resp. are concurrent.
Point of concurrence ?

 

 

[César Lozada]: 

 

1)      Q(P) = R1 ∩ R2 ∩ R3 = midpoint( P, isogonal(P) ), ie., for P=x:y:z (barys)

Q(P) = (2*x+y+z)*a^2*y*z+(b^2*z+c^2*y)*x^2 : :

 

Some ETC-pairs (P,Q(P)): (1,1), (2,597), (3,5), (4,5), (5,8254), (6,597), (7,8255), (8,8256), (9,8257), (10,8258), (11,33562), (13,396), (14,395), (15,396), (16,395), (17, 8259), (18,8260), (20,5894), (21,8261), (23,8262), (24,33563), (25,8263), (31,18805), (32,18806), (35,14526), (36,1737), (40,1158), (54,8254), (55,8255), (56,8256), (57,8257), (58,8258), (59,33562), (61,8259), (62,8260), (64,5894), (65,8261), (67,8262), (68,33563), (69,8263), (75,18805), (76,18806), (79,14526), (80,1737), (84,1158), (399,18285), (484,1749)

 

The locus of P such that Q(P) lies on the Euler line of ABC is K187, thorugh ETCs 3,4,30,74.

 

2)      X(23153)

 

Some others:

 

Q( X(19) ) = MIDPOINT OF X(19) AND X(63)

= a*(a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-(b^2-c^2)^2*a^3+(b+c)*(b^4+c^4-2*(b^2+c^2)*b*c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)*(b-c)*(b^4+c^4)) : : (barys)

= X(18446)-3*X(21160), 3*X(21165)-X(30265)

= lies on these lines: {1, 16599}, {9, 20305}, {19, 27}, {515, 15941}, {2083, 16560}, {4331, 7098}, {5745, 18589}, {6518, 18161}, {7291, 24683}, {16565, 21374}, {18446, 21160}, {18805, 23998}, {21165, 30265}

= midpoint of X(19) and X(63)

= reflection of X(18589) in X(5745)

= [ 2.1971940878418350, 0.4245350740235277, 2.3326660055026290 ]

 

Q( X(22) ) = midpoint of X(22) and X(66)

= (b^2+c^2)*a^12-2*(b^4+c^4)*a^10-(b^4-c^4)*(b^2-c^2)*a^8+4*(b^8+c^8)*a^6-(b^4-c^4)^2*(b^2+c^2)*a^4-2*(b^4-c^4)^2*(b^4+c^4)*a^2+(b^4-c^4)^3*(b^2-c^2): : (barys)

= (2*(7*SW+3*SA)*R^4-(5*SA+8*SW)*SW*R^2+(SA+SW)*SW^2)*S^2+R^2*SB*SC*SW^2 : : (barys)

= lies on these lines: {5, 2781}, {22, 66}, {26, 34118}, {141, 206}, {343, 2393}, {427, 6697}, {1503, 7502}, {3580, 23327}, {9019, 23300}

= midpoint of X(22) and X(66)

= reflection of X(i) in X(j) for these (i,j): (206, 6676), (427, 6697)

= [ -7.3614306312044730, -7.9182756167632520, 12.5201309694533100 ]

 

César Lozada