A Cellular Triangle Containing a Specified Point

 Let P be a set of finite points in the plane in general position, and let x be a point which is not contained in any of the lines passing through at least two points of P. A line l is said to be a k-bisector if both of the two closed half-planes determined by l contain at least k points of P. We show that if any line passing through x is a -bisector and does not contain two or more points of P, then there exist three points P ₁, P ₂, P ₃ of P such that ΔP ₁ P ₂ P ₃ contains x and does not contain points of P in its interior, and such that each of the lines passing through two of them is a -bisector. Received: October 16, 1995 / Revised: October 16, 1996