Problems to discuss in class

1. Prove that the angle we constructed by origami is indeed a trisection of the angle we started with.
2. Prove the "two circle theorem" (If three numbers a, b, c satisfy all possible triangle inequalities (a+b<c, etc) then two circles with centers at distance a, and radious b and c intersect in two points.
3. Consider an isosceles triangle MPQ, with MP and MQ congruent. Through a point A in MQ draw a perpendicular to PQ meeting PQ at point B and meeting the line PM at C. Prove that the triangle MCA is isosceles.