
 AoPS Math Forum :: View topic  orthogonal to the incircle 
  Let Q be the incircle of ABC, and let Q(a), Q(b), Q(c) be circles orthogonal to Q, which pass through B,C ; C,A ; A,B respectively. 
  But now, if you call X, Y, Z the points where the incircle of triangle ABC touches the sides BC, CA, AB, then it is known that the midpoint of the segment YZ is the inverse of the point A in the incircle of triangle ABC. 
  Finally, since the circumcircle of triangle XYZ is the incircle of triangle ABC, we may state that the circumradius of triangle A'B'C' equals 1/2 the radius of the incircle of triangle ABC. 
 www.artofproblemsolving.com /Forum/post14443.html (597 words) 
