BdMO Online Forum

Let,the in-circle of $\Delta ABC$ touches the sides of the triangle at $D,E,F$. Let $X,Y,Z$ be the midpoints of $DE,EF,FD$.Prove that the line joining the circum-centre of $\Delta XYZ$ and the circum-centre of $\Delta ABC$ goes through the in-centre of $\Delta ABC$.

It's actually ISL 14 of 1986. A nice one where I used a little inversion and it worked. Remember, inversion keeps the original centre, the inverted centre and the centre of inversion collinear.