HYACINTHOS 22617

Antreas P. Hatzipolakis

 

[APH]: (Changing notation slightly subscript, not to be confused with
Xi of ETC )

Let ABC be a triangle and IaIbIc the antipedal triangle of P = I
(excentral triangles)

Denote:

Xa, Xb, Xc = the point X(1986) of IaBC, IbCA, IcAB, resp.

Oa,Ob,Oc = the circumcenters of IaBC,IbCA, IcAB, resp.

The circumcircles of XaXbXc, XaBC, XbCA, XcAB are concurrent at a point
Q1

The circumcircles of ABC, AXbXc, BXcXa, CXaXb are concurrent at a point
Q2
[the lines OaXa, ObXb,OcXc concur at Q2].

---- Which points are Q1 , Q2 ?

---- Which point is the center of the circle (Xa,Xb,Xc,Q1) ?

 

 

[Angel Montesdeoca]:

*** First barycentric coordinates:

Q1, (a (a^2 - b^2 + b c - c^2)
(a^2 (b + c) - 2 a b c - b^3 + b^2 c + b c^2 - c^3)) /
((b^2 + c^2 - a^2) (a^6 - a^4 (b^2 - b c + c^2) -
a^3 b c (b + c) - a^2 (b^4 - b^3 c - 2 b^2 c^2 - b c^3 + c^4) +
a b (b - c)^2 c (b + c) + (b - c)^4 (b + c)^2))

with (6-9-13)-search number0.008741569535598741920989990

Q2, a^2/((b - c) (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c + a b^2 c -
a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 + c^4))
with (6-9-13)-search number 5.747003640543809679176906475

Center of the circle (Xa,Xb,Xc,Q1)
a (a^14 (b + c) -
2 a^13 (b + c)^2 - (b - c)^6 (b + c)^5 (b^2 + c^2)^2 +
a^12 (-3 b^3 + 5 b^2 c + 5 b c^2 - 3 c^3) +
2 a^11 (b + c)^2 (4 b^2 - 5 b c + 4 c^2) +
a^10 (b^5 - 21 b^4 c + b^3 c^2 + b^2 c^3 - 21 b c^4 + c^5) -
10 a^9 (b^2 - c^2)^2 (b^2 - b c + c^2) +
a^8 (5 b^7 + 15 b^6 c - 21 b^5 c^2 + 21 b^4 c^3 + 21 b^3 c^4 -
21 b^2 c^5 + 15 b c^6 + 5 c^7) +
2 a^7 b c (-10 b^6 + 7 b^5 c + 4 b^4 c^2 - 18 b^3 c^3 +
4 b^2 c^4 + 7 b c^5 - 10 c^6) +
a^6 (-5 b^9 + 15 b^8 c + 4 b^7 c^2 - 32 b^6 c^3 + 22 b^5 c^4 +
22 b^4 c^5 - 32 b^3 c^6 + 4 b^2 c^7 + 15 b c^8 - 5 c^9) +
2 a^5 (b - c)^2 (5 b^8 + 10 b^7 c - b^6 c^2 + 4 b^5 c^3 +
14 b^4 c^4 + 4 b^3 c^5 - b^2 c^6 + 10 b c^7 + 5 c^8) -
a^4 (b - c)^2 (b^9 + 23 b^8 c + 16 b^7 c^2 + 28 b^5 c^4 +
28 b^4 c^5 + 16 b^2 c^7 + 23 b c^8 + c^9) -
2 a^3 (b^2 - c^2)^2 (4 b^8 - 7 b^7 c + b^6 c^2 + 5 b^5 c^3 -
12 b^4 c^4 + 5 b^3 c^5 + b^2 c^6 - 7 b c^7 + 4 c^8) +
a^2 (b - c)^4 (b + c)^3 (3 b^6 + 8 b^5 c - 4 b^4 c^2 +
18 b^3 c^3 - 4 b^2 c^4 + 8 b c^5 + 3 c^6) +
2 a (b^2 - c^2)^4 (b^6 - 3 b^5 c + 4 b^4 c^2 - 6 b^3 c^3 +
4 b^2 c^4 - 3 b c^5 + c^6))

with (6-9-13)-search number 3.12452916619761654679673584

Best regards
Angel Montesdeoca