HYACINTHOS 29324

[Antreas P. Hatzipolakis]:

 

 

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

 

Denote:

 

(Oa), (Ob), (Oc) = the circles with diameters BC, CA, AB, resp.

(N) = the pedal circle of O = the NPC of ABC

 

Ra = the radical axis of (Oa), (N)

Rb = the radical axis of (Ob), (N)

Rc = the radical axis of (Oc), (N)

 

A* = Ra /\ B'C'

B* = Rb /\ C'A'

C* = Rc /\ A'B'

 

1. ABC, A*B*C* are perspective

(AA*, BB*, CC* are perpendiculars to Euler line)

 

2. A'B'C', A*B*C* are perspective (at a point on the NPC)

 

3. The circumcircles of A*BC, B*CA, C*AB are concurrent at U

[U = the Miguel point of A*B*C* wrt A, B, C, since A, B, C lie on the sidelines  B*C*, C*A*, A*B* of A*B*C*, resp.]

 

4. The circumcircles of A'B*C*, B'C*A*, C'A*B* are concurrent at W

[W = the Miguel point of A'B'C' wrt A*, B*, C*, since A*, B*, C* lie on the sidelines B'C', C'A', A'B' of A'B'C', resp.]

 

 

[Ercole Suppa]:

 

1. Perspector(ABC, A*B*C*) = X(523)

 

2. Perspector(A'B'C', A*B*C*) = X(115)

 

3. U = X(6328) 

 

4. W = COMPLEMENT OF X(14366) =

 

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

 

= 3*X[2]-X[14366]

 

= lies on these lines: {2,14366}, {3,30715}, {523,6328}, {868,5489}, {3767,6792}, {5007,23991}

 

= complement of X(14366)

 

= (6-9-13)  search numbers:  [4.1138881746676335993, 2.8238545687674574984, -0.2130293770858454416]

 

 

Best regards,

Ercole Suppa