In June 2003, I think it will be time to use a new rule for naming polygons. Are you ready to hear it??
The first change is that the word "triangle" will be dropped completely from our language. The 2 new words will be "trigon", for equilateral triangles, and "trilateral", for any other triangle.
Also, there will be 2 new terms for special polygons with any number of sides:
Regular: A regular polygon is a polygon with all sides congruent and all angles congruent.
Semiregular: A semiregular polygon is a polygon that is not regular, but whose vertexes made by its angles all lie on one circle of a given size.
Note that using these definitions, triangles are always either regular or semiregular. In other words, a regular 3-sided figure will be called a trigon and any other 3-sided figure will be called a trilateral. Polygons with 4 or more sides, however, can be regular, semiregular, or neither. For naming 4-sided polygons, we will use all the same terms used earlier, but change the definitions as follows:
Kite: A quadrilateral with 2 pairs of consecutive congruent sides, but opposite sides not NECESSARILY congruent.
Trapezoid: A quadrilateral with AT LEAST one pair of parallel sides.
Isosceles trapezoid: A trapezoid whose BASE ANGLES are congruent.
Parallelogram: A trapezoid whose opposite angles are congruent.
Also, note the following theorem:
A trapezoid is also a semiregular quadrilateral if and only if it is an isosceles trapezoid but not a square. (A square is the regular quadrilateral.) So, we now have a new acceptable definition of an isosceles trapezoid: a quadrilateral that is both a trapezoid and either regular or semiregular. Also, we can extend this theorem to saying that a quadrilateral is semiregular if and only if its opposite angles add up to 180 degrees but is not a square and that a quadrilateral is regular if and only if it is a square.
So now we have trigon and square for regular polygons with 3 and 4 sides, respecively, and trilateral and semiregular quadrilateral for semiregular polygons with 3 and 4 sideds, respectively. Also, a quadrilateral that is neither regular nor semiregular will be called a "4-side". The word "quadrilateral" will be the only word retained in our language that can be used to refer to polygons with a given number (4) of sides that can be regular, semiregular, or neither.
The terms "pentagon", "quintilateral", and "5-side" will be used for 5-sided figures that are regular, semiregular, and neither, respectively. The terms "hexagon", "sextilateral", and "6-side" will be used for 6-sided figures that are regular, semiregular, and neither, respectively.
Now, let me go from 3 to 12 sides, and make a list of all the acceptable names for polygons that are regular (all sides are congruent and all angles are congruent,) semiregular (vertexes all lie on a circle but sides are not all congruent,) and neither.
polygon: any number of sides quadrilateral: 4 sides, can be regular, semiregular, or neither kite: quadrilateral with 2 pairs of consecutive congruent sides trapezoid: quadrilateral with at least one pair of parallel sides isosceles trapezoid: trapezoid with base angles congruent parallelogram: trapezoid with opposite angles congruent rhombus: quadrilateral that is both a kite and a parallelogram rectangle: trapezoid that is both an isosceles trapezoid and a parallelogram
Note that "3-side" is not included because all 3-sided polygons are either regular or semiregular.
That's it! Now we have 3 types of polygons (regular, semiregular, neither) and 3 languages (Greek, Latin, English,) so we can think of one as corresponding to the other in the following way. Can you PLEASE reply to this message, telling me if you have any comments??