Level 5: Construct a line (segment) perpendicular to line segment AB going through point C.

Hint: Perpendicular bisector of a chord go through the center.

Solution: Without any particular reason, let construct two circles with center A and center B with radius AC and BC respectively. The two circle intersect at C and we name the other intersection E. Name the intersection between CE and AB as F. Now CF = EF because of symmetry. Triangle BCF is congruent to triangle BEF, so angle CFB = angle EFB, but they must also sum to 180 degree, so it is a right angle and we are done.