Parahexes are hexagons that share some properties of a parallelogram. (I thought that "parahexus" was a shorter and better name than "parallelohexagon.")

Definition: a parahexus is a hexagon with all three pairs of opposite sides both parallel and congruent.

The word "both" in this definition is important. In a hexagon opposite sides being congruent does not imply that they are parallel and parallel opposite sides need not be congruent. For example, take an equilateral triangle with smaller equilateral triangles lopped off the corners. While opposite sides are parallel, such a figure is not a parahexus.

In order to prove that a hexagon is a parahexus, one does not have to prove all pairs of opposite sides both parallel and congruent. In fact, as an extension of the methods used for parallelograms, you can prove that the hexagon is a parahexus by proving any one of the following:

All pairs of opposite sides are parallel and one pair of opposite sides is congruent.	All pairs of opposite sides are congruent and one pair of opposite sides is parallel.	Two pairs of opposite sides are both parallel and congruent.

In all three cases the remaining parallels or congruences can be verified.

Theorem 1: Other ways to prove that a hexagon is a parahexus are also possible. If the 3 main diagonals of a hexagon are concurrent, and the point of concurrence is the midpoint of each diagonal, then the hexagon is a parahexus.

Justification: Using SAS and the fact that vertical angles are congruent you can prove the triangles opposite each other across the point of concurrence are congruent and thus prove opposite sides congruent, and prove them parallel using the congruent alternate interior angles. This is very much like the "diagonals bisect each other" method for proving that a quadrilateral is a parallelogram.

Theorem 2: A hexagon formed by a parallelogram with congruent triangles placed on one pair of opposite sides of the parallelogram so that opposite sides of the resulting hexagon are congruent forms a parahexus.

Justification: One pair of opposite sides are already parallel and congruent, and the other 2 pairs are congruent because they are corresponding parts of congruent triangles.

Properties of a parahexus:

• Each pair of opposite angles are congruent.
• Any pair of diagonals that do not intersect inside or on the parahexus are parallel.
• The diagonals connecting the endpoints of opposite sides, along with the sides themselves, form a parallelogram.
• A triangle whose vertices are consecutive vertices of a parahexus is congruent to the triangle determined by the three remaining points of the parahexus.
• The three main diagonals bisect one another.
• Using the congruent triangles, one can prove that any one main diagonal bisects any other main diagonal. A corollary is that the diagonals of a parahexes are concurrent.
• Triangles determined by alternating vertices are congruent.

These properties can be verified using the parallel and congruent sides to establish many congruent triangles. Be sure to observe the many parallelograms formed by the sides and diagonals of the parahexus.

If the parahexus is warped so that it is no longer a convex figure (or even if the sides are "pulled through" each other, leaving a six-sided non-polygonal arrangement), some diagonals will go outside the figure, but all the properties will hold.

Most of the parahexus properties can be extended to further polygons. Any n-gon in which n is even can be a parallelopolygon. In all cases, you can prove congruent triangles are formed by the sides and the main diagonals and that the main diagonals both bisect each other and are concurrent.

Parahexes do have a useful property. (What? These things can actually be used for something?) Any parahexus can be used to tessellate a plane. Given any parahexus, one can place six identical ones, oriented in the same direction, on each side of it. These will match up perfectly because pairs of opposite sides are congruent and the three angles around the vertex where the parahexes meet will add up perfectly to 360º. This is a consequence of the fact that opposite angles are congruent and the sum of the angles is 720º. More parahexes can be added to the edges of these, and more to those, on to infinity. Interesting shaped parahexes could be used as floor tiles, being almost as easy to make and a lot better looking than plain old squares or regular hexagons.

Also, it is appears that no hexagon except a parahexus can infinitely tessellate a plane by simple translation. If pairs of opposite sides of a given hexagon are not all congruent, then the sides would not fit together and gaps would be left. This is rather obvious. And if opposite sides are not parallel, three adjacent angles (which would be the same as the three angles around a point where three hexagons meet) would not add up to 360º. This means all opposite sides must be parallel and congruent, therefore, a parahexus is necessary. Concave parahexes will also work for tessellating a plane.

This feature of tessellating a plane does not extend to parallelo-8-gons, ||-10-gons, ||-12-gons, ||-50-gons, etc.

Additional Questions:

1. Find some other concise methods to prove that a hexagon is a parahexus.
2. Prove that a figure formed by connecting alternate vertices of a paraoctagon is a parallelogram.
3. What special features does a paradecagon have?
4. What minimal conditions are necessary for two parahexes to be congruent?
5. What minimal conditions are necessary for two parahexes to be similar?
   
   
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