ADGEOM 1388

 

[Tran Quang Hung]:

Dear geometers,

Let ABC be a triangle with nine-points circle (N) and d is a tangent of (N). Let da,db,dc be reflections of d through BC,CA,AB, resp. Lines da,db,dc form a triangle XYZ. I see that circumcircle of triangle XYZ is always touches a fixed circle internally when d moves.

Could you please help me, Is it known circles ?

Best regards,

Tran Quang Hung.

 

[César Lozada]:

 

Dear Tran Quang Hung:

It is the circle with diameter (X(3), X(1614)), with center

    Z = (3*SA^2-2*(3*SW-5*R^2)*SA+S^2)*a : :  (trilinears)

       =  Midpoint of (3,1614)

       =  On lines (3,74), (49,1154), (52,1493), (54,143), (184,1658), (546,1495), (567,3518), (3146,3431)    

       =  ( 0.581261729932052, -1.27818900697439, 4.257289919074934 ) 

Its radius is

   Radius = (R/2)*(3+cc)/(1+cc)

   Where cc=2*(cos(2*A)+cos(2*B)+cos(2*C))

 

Best regards

César Lozada