HYACINTHOS 26454

[Adrienmath] (*)

 

Let ABC be a triangle and A'B'C' the cevian triangle of the symmedian point X(6).

The parallel through A' to AB intersects AC at Ab.
The parallel through A' to AC intersects AB at Ac.

B, C, Ab, Ac are concyclic. Let Oa be the center of the circle.

Similarly Ob, Oc.

The triangles ABC, OaObOc are perspective.

Perspector ?

 

(*) Here

 

 

[ Angel Montesdeoca]:

 

 

*** The perspector of triangles ABC, OaObOc is X(182), midpoint of Brocard  diameter.

*** The fixed  point of the affine transformation between the two triangles  ABC and OaObOc is 
        F = (a^2 (b^2+c^2) (-a^4+2 b^2 c^2+a^2 (b^2+c^2)):...:...),
the intersection of  Euler line and the line that  passes through the symmedian and the perspector, X(11175), of half-Moses circle.

F lies on lines X(i)X(j) for these {i, j}: {2, 3}, {6, 11175}, {32, 10014}, {39, 3051}, {51, 5188}, {95, 6394}, {141, 1634}, {160, 3763}, {216, 9475}, {263, 1350}, {574, 3117}, {592, 3398}, {597, 5201}, {669, 11183}, {1078, 3978}, {1613, 5013}, {1899, 7800}, {2076, 8570}, {2979, 3095}, {3060, 9821}, {3589, 8266}, {3819, 13334}, {5012, 12054}, {5024, 9463}, {5092, 5191}, {5106, 8589}, {5650, 9155}, {7767, 11245}, {7998, 11171}.

 (6 - 9 - 13) - search numbers  of F : (3.97779631259842, 3.09778286259012, -0.339860413392666).
 
 Angel Montesdeoca