HYACINTHOS 25879

[‎Tran Quang Hung‎]:

 

Let ABC be a triangle with incircle (I). Circle (wa) passes through B,C and is tangent to (I). Similarly, we have circle (wb),(wc). Circle (wb),(wc) intersect again at A'. Similarly, we have B',C'. The circumcenter of triangle A'B'C' lies on OI line of triangle ABC.
Point?

Reference: Romantics of Geometry #708.

 

[Peter Moses]:

Hi Antreas,

 

X(9940).

 

Best regards,

Peter Moses.

 

[APH]:

Which point is it wrt the pedal triangle of I ? (lying on the Euler line of the pedal triangle of I).

That is, tangential triangle version:

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

Let Oa be the circle tangent to the circumcircle and passing through B' and C'. Similarly Ob and Oc.
Let A* be the intersection, other than A', of circles Ob and Oc. Similarly B* and C* 
The circumcenter of A*B*C* lies on the Euler line of ABC.


[Peter Moses]:

Hi Antreas,

 

>Which point is it wrt the pedal triangle of I ?

 

2 a^10-5 a^8 b^2+2 a^6 b^4+4 a^4 b^6-4 a^2 b^8+b^10-5 a^8 c^2+8 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-8 a^2 b^4 c^4+2 b^6 c^4+4 a^4 c^6+8 a^2 b^2 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10:: 

 

on lines {{2,3},{68,154},{156,206},{ 343,10539},{498,9645},{511, 9820},...}.

 

midpoint of X(i) and X(j) for these {i,j}: {{5,26},{2883,7689},{6759, 12359},{10154,10201}}.

 

reflection of X(i) in X(j) for these {i,j}: {{140,10020},{11250,3530}}.

 

3 (3 J^2 - 7) X[2] + (13 - J^2) X[4]. (J = OH/R)

 

Searches: {-0.78195214642148495880,-1. 6494092598128171095,5. 1434641908954763138}.

 

Best regards,

Peter Moses.