Question 47
In the plane, the points $A, B, C, D, E, F$ satisfy $\triangle BCD \stackrel{+}{\sim} \triangle ECA \stackrel{+}{\sim} \triangle BFA$ (where $\stackrel{+}{\sim}$ denotes direct similarity), and $I$ is the incenter of $\triangle ABC$.
Prove that the circumcenters of the three triangles $AID, BIE, CIF$ are collinear.
Note: two triangles $\triangle ABC \stackrel{+}{\sim} \triangle DEF$ are directly similar if, besides the two triangles being similar, the direction of rotation from $A$ to $B$ to $C$ is the same as that from $D$ to $E$ to $F$, i.e. both are clockwise or both are counterclockwise.
Prove that the circumcenters of the three triangles $AID, BIE, CIF$ are collinear.
Note: two triangles $\triangle ABC \stackrel{+}{\sim} \triangle DEF$ are directly similar if, besides the two triangles being similar, the direction of rotation from $A$ to $B$ to $C$ is the same as that from $D$ to $E$ to $F$, i.e. both are clockwise or both are counterclockwise.