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In mathematics, there are terms with no consensus about their definitions:
different people use the same mathematical term with different
meanings. This is the case of the term “polygon”. In the software of this activity,
we use the definitions given below. They can be different from the definitions used in
the book you're studying.
Definition (Polygonal Line).
A polygonal line is a planar figure formed by a sequence of points
A1, A2, ..., An
and the segments
A1A2, A2A3, ..., An−1An.
The points are the vertices of the polygonal line and the segments are their sides.
If An = A1, we say that the polygonal line is closed.
Definition (Polygon).
A polygon is a polygonal line
with the following properties:
(a) it is closed,
(b) each one of its vertices is an end of two sides and
(c) two sides with a common end are not collinear.
Definition (Simple Polygon).
A polygon P is simple provided the only points of
the plane that belong to two edges of P are the vertices of P.
Simple polygons also are called of Jordan's polygons, because
the Jordan's curve theorem
can be used to demonstrate that such a polygon divides the plan in two regions,
the interior region inside of it and the exterior region outside of it.
Definition (Convex Polygons).
We say that a simple polygon is convex if its interior C is convex, i.e.,
when any line segment
joining two points of C is completely contained in C. A convex
polygon is always contained in one of the half-planes determined by the straight lines
that contain its sides.
Definition (Quadrilateral).
A quadrilateral is a polygon with four sides.
Definition (Parallelogram).
A parallelogram is a convex quadrilateral whose opposite sides are parallels and congruents.
Definition (Rhombus).
A rhombus is a parallelogram which all sides are congruents.
Definition (Kite).
A kite is a convex quadrilateral with two adjacent sides having the same length, say a,
and the other two sides (adjacents) also having the same length, say b.
The rhombus is a special case of kite (where a = b).
Definition (Rectangle).
A rectangle is a convex quadrilateral with four right angles.
Definition (Square).
A square is a rectangle that also is a rhombus, in other words, a square
is a convex quadrilateral with four right angles and four congruent sides.
Definition (Trapezium).
A trapezium is a convex quadrilateral with parallel opposite sides.
The parallel opposite sides are called bases and the other two are called laterals.
A trapezium is called isosceles if its laterals are congruents.
A trapezium is called scalene if its laterals are not congruents.
A trapezium is called right if it has at least two right angles.
Definition (Diagonal of a Polygon).
A diagonal of a polygon is any straight line segment
joining two non consecutive vertices of the polygon.
Definition (Orthodiagonal Quadrilateral).
A quadrilateral is orthodiagonal if its two diagonals are perpendicular.
Definition (Cyclic Polygon).
A cyclic polygon is a polygon with vertices upon which a circle can be circumscribed.
A polygon is cyclic if all its vertices belong to the same circle, that is,
if there is a circle which circumscribe the polygon.
Definition (Inscriptible Polygon).
A polygon is inscriptible if its sides are tangent to a same circle.
In this situation, we say that the polygon circumscribes the circle.
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