2011 IMO Problems/Problem 6
Let be an acute triangle with circumcircle . Let be a tangent line to , and let and be the lines obtained by reflecting in the lines , and , respectively. Show that the circumcircle of the triangle determined by the lines and is tangent to the circle .
Without loss of generality, let be the unit circle and let be the line .
Denote the coordinates of by and similarly for B and C.
We get , ...
The equation for line is
Through a little bash, (0,1) reflects to . simplifies to . The terms for the other 2 are symmetric. The intersection point must reflect to itself, and the equation is .
It is trivial to find the intersections of a,b and their perpendicular bisectors, so this is left to the reader as an exercise.
Regardless, the circumcenter and an intersection of the circles are collinear with (0,0), so it is a tangency.