Question 153
Let $k$ be a positive real number and $ABC$ is an acute triangle which is not isosceles. Denote by $O$ the circumcenter and let $AD$, $BE$ and $CF$ be its inner angle-bisectors. Choose the points $L$, $M$ and $N$ on $AD$, $BE$ and $CF$ respectively such that $\frac{AL}{AD} = \frac{BM}{BE} = \frac{CN}{CF} = k$. Denote by $(O_1)$, $(O_2)$, and $(O_3)$ the circles that, respectively, passing through $L$ and touch $OA$ at $A$, passing through $M$ and touch $OB$ at $B$, passing through $N$ and touch $OC$ at $C$.
1/ Prove that when $k = \frac{1}{2}$, the three circles $(O_1)$, $(O_2)$ and $(O_3)$ have exactly two common points. Furthermore, the centroid $G$ of the triangle $ABC$ is on the line connecting these two common points.
2/ Find all values of $k$ such that the three circles $(O_1)$, $(O_2)$ and $(O_3)$ have exactly two common points.
1/ Prove that when $k = \frac{1}{2}$, the three circles $(O_1)$, $(O_2)$ and $(O_3)$ have exactly two common points. Furthermore, the centroid $G$ of the triangle $ABC$ is on the line connecting these two common points.
2/ Find all values of $k$ such that the three circles $(O_1)$, $(O_2)$ and $(O_3)$ have exactly two common points.