L

A

C H

B

B

12

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 .

CA CB

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

A

O

O

O OL

O O

AA A

A

C

B

N

L

M

1O 2O

3O

1I 2I

3I

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

ABC 1O 2O 3OALN BLM CMN

1I 2I 3I.321321 IIIOOO

Problem. Given three parallel lines, find an equilateral triangle whose vertices lie on them.

L

M

A

M

M

CB

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

B C

1O

2O

3O3OA

B C

ABCABC

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

K

K

P

K