VN-TST 2008 Problem 5

Let $k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $\frac {AL}{AD} = \frac {BM}{BE} = \frac {CN}{CF} = k$ . Denote $(O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C.

1. Prove that when $k = \frac{1}{2}$ , three circles $(O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles.
2. Find all values of k such that three circles $(O_1),(O_2),(O_3)$ have exactly two common points