1. L A C H B 1 2 Problem. Given two points A, B on the same side of line Find the point C on L such that and make congruent angles with L.

2. Problem. Point Q is called a center of symmetry for figure F if whenever is a segment having Q as midpoint and A in F, then also belongs to F. Show that a figure can only have zero, one, or infinitely many centers of symmetry. A O O L

3. A C B N L M Problem. Let L, M, N be the respective midpoints of sides AB, BC, CA of. Let,, be the circumcenters of triangles,, respectively, and let,, be the incenters of these same triangles. Show that

4. Problem. Given three parallel lines, find an equilateral triangle whose vertices lie on them. L M A C B

5. Problem. For any triangle, construct equilateral triangles on the sides of, exterior to it. Show that the centers of these triangles also form the vertices of an equilateral triangle. A BC A BC

6. Problem. Given a circle K and a point P on K. Find the locus of midpoints M of all chords PA of K through P. A P