ILL 1990 Problem 28

Let $ABC$ be an arbitrary acute triangle. Circle $\Gamma$ satisfies the following conditions: (i) Circle $\Gamma$ intersects all three sides of triangle $ABC.$ (ii) In the convex hexagon formed by above six intersections, the three pairs of opposite sides are parallel respectively. (The hexagon maybe degenerate, that is, two or more vertices are coincide. In this case, "opposite sides are parallel" is defined through limit opinion.) Find the locus of the center of circle $\Gamma$ , and explain how to construct the locus.