Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares. 

Now, a polygon is equiangular if all of its internal angles are congruent. 

Some important facts to consider

  • The only equiangular triangle is the equilateral triangle
  • If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon. 
  • Rectangles, including squares, are the only equiangular quadrilaterals
  • Equiangular polygon theorem. Each angle of an equiangular n-gon is 
$$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$

Viviani's theorem 
Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in mathematics, Viviani became a pupil and associate of Galileo (1564-1642). Viviani’s Theorem is one of his widely known achievements. This theorem is about equilateral triangles and its states that inside an equilateral triangle, the sum of the perpendicular distances from a point P to the three sides is independent of the position of P (and so equals the altitude of the triangle). In other words, For any given point inside an equilateral triangle, the sum of the perpendicular distances between the point and the three sides is always a fixed value.

This theorem can also be extended to equilateral and equiangular polygons that states, The sum of distances from the lines containing the sides of a convex equilateral or equiangular polygon to any point in its interior does not depend on the point. 

Some problems to consider - 

1. All angles of a polygon are equal to $180^{\circ}$. How many sides does the polygon have?
2. All angles of a polygon are equal to $k$, where $k$ is a whole number. How many different values are possible for $k$?
3. Is there a equiangular octagon with perimeter $20$ all of whose side lengths are among the numbers $1, 2, 3,4$?

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