HYACINTHOS 29499

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle, P a point and A'B'C', A"B"C" the cevian, circumcevian triangles of P, resp.

Denote:

(Oa), (O1) = the circles with diameters BC, A'A", resp.
(Ob), (O2) = the circles with diameters CA, B'B", resp.
(Oc), (O3) = the circles with diameters AB, C'C", resp.

Ra = the radical axis of (Oa), (O1)
Rb = the radical axis of (Ob), (O2)
Rc = the radical axis of (Oc), (O3)

A*B*C* = the triangle bounded by Ra, Rb, Rc.

1. For P = H:
ABC, A*B*C* are perspective.
Perspector = X(34285)
See also HG140919

I think that for:

2. P = O
3. P = I

The triangles ABC, A*B*C* are also perspective.  

 

[Peter Moses]:

Hi Antreas,

2). X(3)X(1075) ∩ X(97)X(3164) =

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

= lies on the circumconics {{A,B,C,X(2),X(3)}}, {{A,B,C,X(4),X(8613)}}, {{A,B,C,X(64),X(8794)}}, {{A,B,C,X(92),X(6360)}}, {{A,B,C,X(253),X(8795)}}, {{A,B,C,X(275),X(1294)}}, {{A,B,C,X(324),X(3164)}}, {{A,B,C,X(393),X(17849)}}, {{A,B,C,X(459),X(15412)}}, the curve Q124 and these lines: {3, 1075}, {97, 3164}, {394, 8613}, {14941, 34186}
.
= polar conjugate of X(1075)
= cyclocevian conjugate of X(68)
= isotomic conjugate of the anticomplement of X(2052)
= polar conjugate of the isogonal conjugate of X(13855)
= X(13855)-anticomplementary conjugate of X(21270)
= X(i)-cross conjugate of X(j) for these (i,j): {2052, 2}, {15318, 253}
= cevapoint of X(216) and X(13322)
= barycentric product X(264)*X(13855)
= barycentric quotient X(i)/X(j) for these {i,j}: {4, 1075}, {13855, 3}

3).
X(15446)

Points, P, on Darboux cubic K004 are perspective too but are generally nasty.

Best regards,
Peter Moses.