three circles
Source: Vietnam TST 2008, Problem 5
September 5, 2008
geometrycircumcircleEulerpower of a pointradical axisgeometry proposed
Problem Statement
Let 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} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k. Denote 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 \equal{} \frac{1}{2}, three circles 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 have exactly two common points