Construct the tangent thru a point on the circle: http://youtu.be/7PdAkCaeI2s
Construct the tangent thru a point outside the circle: http://youtu.be/YtGK1cA4EGI
Square inscribed in a circle: https://www.youtube.com/watch?v=2gNfltBkbkI
Circle through three noncollinear points (same as circumcenter construction!): https://www.youtube.com/watch?v=i4DXA7T7m_w