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 math...
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Problems - 1

Let's look at some interesting applications of the basic arithmetic formulas in the problems below.

 Problem 1 - Evaluate $123456789 \cdot 999999999$.

       Solution - $123456789 \cdot 999999999 = 123456789 \cdot (1000000000-1)$
                        $=123456789000000000-123456789=\boxed{123456788876543211}$

Problem 2 - Find the value of $\frac{13579}{(-13579)^2+(-13578)(13580)}$.

       Solution - Using $(a-b)(a+b) = a^2 - b^2$, we have
$$\frac{13579}{(-13579)^2+(-13578)(13580)} = \frac{13579}{(13579)^2-(13579^2-1)} = \boxed{13579}$$

Problem 3 - Evaluate $\frac{83^3+17^3}{83 \cdot 66+17^2}$.

       Solution - By the use of the formula $a^3+b^3=(a+b)(a^2-ab+b^2)$
$$\frac{83^3+17^3}{83 \cdot 66+17^2} = \frac{(83+17)(83^2-83 \cdot 17 + 17^2)}{83 \cdot 66 + 17^2} = \frac{100 \cdot (83 \cdot 66 + 17^2)}{83 \cdot 66 + 17^2} = \boxed{100}$$

Problem 4 - Evaluate
$$\frac{(4 \cdot 7 + 2)(6 \cdot 9 + 2)(8 \cdot 11 + 2) ... (100 \cdot 103 + 2)}{(5 \cdot 8 + 2)(7 \cdot 10 + 2)(9 \cdot 12 + 2) ... (99 \cdot 102 + 2)}$$.

       Solution - Notice $n(n+3)+2 = n^2+3n+2 = (n+1)(n+2)$ for any integer $n$. Using this, we have

$$\frac{(4 \cdot 7 + 2)(6 \cdot 9 + 2)(8 \cdot 11 + 2) ... (100 \cdot 103 + 2)}{(5 \cdot 8 + 2)(7 \cdot 10 + 2)(9 \cdot 12 + 2) ... (99 \cdot 102 + 2)} = \frac{(5 \cdot 6)(7 \cdot 8)(9 \cdot 10) ... (101 \cdot 102)}{(6 \cdot 7)(8 \cdot 9)(10 \cdot 11) ... (100 \cdot 101)} = 5 \cdot 102 = \boxed{510}$$

Problem 5 - Evaluate $\frac{20092008^2}{20092007^2+20092009^2-1}$.

       Solution - $$\frac{20092008^2}{20092007^2+20092009^2-1} = \frac{20092008^2}{20092007^2-1) + (20092009^2-1)} = \frac{20092008^2}{(20092006)(20092008)+(20092008)(20092010)} = \frac{20092008^2}{(20092008)(20092006+20092010)}=\frac{20092008^2}{2(20092008^2)} = \boxed{\frac{1}{2}}$$

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Equilateral and Equiangular Polygons

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