1.4. Polygons and Triangles, Kiselev's Geometry, Book I. Planimetry, Givental Alexander.

 Some notes.

1. If a broken line is drawn such that the segment joining any two points of the broken line does not intersect the broken line, then the broken line is called closed.

- The figure above though is a counterexample.

2. For a broken line to be closed it is not necessary for it to be convex. Nor is it necessary for the broken line to be convex, that it be closed.

3. A segment joining two vertices of a polygon that have at least one point lying between them is called a diagonal of the polygon.

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 Exercises.

55. Prove that each diagonal either lies entirely in its interior, or entirely in its exterior.

Proof: Assume that a diagonal of a quadrilateral lies both in its interior and its exterior. This implies that the diagonal passes through a side of the quadrilateral The side connects two vertices. The diagonal will connect two opposite vertices. The resulting broken line ( which forms the boundary of the polygon ), though is not closed. A quadrilateral has to be closed. This implies that the polygon is not a quadrilateral. We have been so led to a contradiction.

56. Show that in any triangle, every two medians intersect.

Proof: Suppose that two medians of a triangle do not intersect. In other words, these two medians are parallel to each other. The medians join the midpoint of a side to its opposite vertex. There are two medians, so we should have two sides and two vertices. The resulting broken line which forms the boundary of the polygon though is not closed. A triangle has to be closed. The polygon so formed cannot be a triangle. A contraction has been so arrived at.

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