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Plane Separation

Definition A set in is called convex provided it has the property that for all points and the segment joining and lies in that is,

(H-1) Let be any line lying in any plane The set of all points in not on consists of the union of two subsets and of such that

     (i) and are convex sets
     
     (ii) and have no points in common
     
     (iii) If lies in and lies in the line intersects the segment
     

Theorem (22) If holds and passes through point but not point then and lie on opposite sides of line

Theorem (23) If point lies on and point lies in one of the half planes determined by then, except for the entire segment or ray lies in that half-plane.

Theorem (24) Let and lie on opposite sides of a line and let and be any two distinct points on Then the segment and ray have no point in common.

Theorem (25) Suppose and are any three distinct noncollinear points in a plane, and is any line in that plane that passes through an interior point of one of the sides, of the triangle determined by and Then line meets either at some interior point the cases being mutually exclusive.

Definition The interior of an angle is the set of all points that simultaneously lie on the of and on the of

Theorem (26) If and lie on the sides of then, except for the end points, segment is a subset of the interior of If Interior then, except for ray

Theorem (27) If lies in the interior of then ray meets segment at some interior point

Geometry Books

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