Question 5
Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$. Let $AD,BE,CF$ be its internal angle bisectors. Choose points $L,M,N$ on the lines $AD,BE,CF$, respectively, such that
$$\frac{AL}{AD}=\frac{BM}{BE}=\frac{CN}{CF}=k,$$
where $k$ is a positive constant. Let $(O_1),(O_2),(O_3)$ be the circles which respectively pass through $L$ and are tangent to $OA$ at $A$; pass through $M$ and are tangent to $OB$ at $B$; and pass through $N$ and are tangent to $OC$ at $C$.
1. Prove that for $k=\frac12$ the three circles have exactly two common points, and the line joining these points passes through the centroid of $ABC$.
2. Find all values of $k$ for which the three circles have exactly two common points.
$$\frac{AL}{AD}=\frac{BM}{BE}=\frac{CN}{CF}=k,$$
where $k$ is a positive constant. Let $(O_1),(O_2),(O_3)$ be the circles which respectively pass through $L$ and are tangent to $OA$ at $A$; pass through $M$ and are tangent to $OB$ at $B$; and pass through $N$ and are tangent to $OC$ at $C$.
1. Prove that for $k=\frac12$ the three circles have exactly two common points, and the line joining these points passes through the centroid of $ABC$.
2. Find all values of $k$ for which the three circles have exactly two common points.