HYACINTHOS 28926

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C', A"B"C" the pedal, antipedal triangles of O, resp. (medial, tangential triangles)

Denote:

Ba, Ca = the orthogonal projections of B, C on B"C", resp.
(Ob1), (Oc1) = the circumcircles of ABaC', ACaB', resp.

Similarly:

Cb, Ab = the orthogonal projections of C, A on C"A", resp.
(Oc2), (Oa2) = the circumcircles of BCbA', BAbC', resp.

Ac, Bc = the orthogonal projections of A, B on A"B", resp.
(Oa3), (Ob3) = the circumcircles of CAcB', CBcA', resp.  

The six circles (Ob1), (Oc1), (Oc2), (Oa2), (Oa3), (Ob3) are equal to NPC (N).

Reference: Ioannis Panakis (in a 1961 paper in Greek)

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Now, the six circumcenters Ob1, Oc1, Oc2, Oa2, Oa3, Ob3 lie on a conic centered at N
(for Oa2Oa3, Ob3Ob1,  Oc1Oc2 have common midpoint the N)

Perspector of the conic ?

[Angel Montesdeoca]:
 

Perspector of the conic:

W =  X(324)X(547)∩X(343)X(15699)

= 1/(5 a^8-17 a^6 (b^2+c^2)+a^4 (21 b^4+17 b^2 c^2+21 c^4)-11 a^2 (b^2-c^2)^2 (b^2+c^2)+(b^2-c^2)^2 (2 b^4-7 b^2 c^2+2 c^4)) :  :

= lies on these lines: {324,547}, {343,15699}, {5055,31610}.
 
 (6 - 9 - 13) - search numbers  of W: (3.73940619993519, 1.12729966185518, 1.13434646988378).
 
 Angel Montesdeoca