Let ABC be a triangle in the plane. Find circles C0;C1; : : : ;C6 such that Cj

Question

has exactly j points in common with the boundary of ABC (this \boundary"
consists of the line segments AB, BC, CA). Is it possible to �find a circle C7
with 7 points in common with the boundary of ABC?

Answer 1

Of course not. Each line can intersect any circle in at most two points. The three lines can thus intersect a circle in at most 6 points.