Question

Question #5ab10

Answer 1

See explanation below.

Explanation:

We have to start from basic undefined objects of Geometry (point, straight line, plane) that satisfy basic axioms.

Then we will define a concept of half-plane and, based on this definition, it will be obvious that half-plane is not empty.

At this point it is important to understand the basic principles of Geometry, including undefined objects, relationship between them and axiom these objects must satisfy. A good introduction to this can be found on UNIZOR.COM in the chapter Geometry, topic Elements of Plane Geometry.

The Web gives not very rigorous explanation of half-plane:
A half-plane is a planar region consisting of all points on one side of an infinite straight line, and no points on the other side. If the points on the line are included, then it is called an closed half-plane.

We might define it a little more rigorously as follows.
Consider a plane `delta`. It's a non-empty set of points.
Consider a straight line `l` on this plane, also non-empty set of points.
This straight line is a subset (`l sub delta`) of a plane, that is not the same as a plane. Therefore, there exists a point `A` that belongs to a plane (`A in delta`) but does not belong to a straight line (`A notin l`).

We also need a concept of a segment - a subset of a straight line in between two points on it. Any two points on a plane can be connected by a segment.

Consider all points on plane `delta` that can be connected to point `A` (that lies outside of straight line `l`) by a segment not intersecting with line `l`. It's not an empty set since it contains, at least, point `A` itself. This subset of plane `delta` we will call a half-plane defined by plane `delta`, straight line `l` on it and point `A` lying on the plane `delta` but not on line `l`.

Such an approach guarantees that our half-plane is non-empty since it contains, at least, one point `A`.

So, non-emptiness of half-plane is a direct consequence of its definition. More rigorous definition leads to non-emptiness relatively straight forward.