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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Common Factorizations

Factorization is the process of writing a number or expression as a product of several factors, usually smaller or simpler. For example $8$ can be factorized as $2 \cdot 4$, and $(x-3)(x+3)$ is the factorization of the polynomial $x^2-9$. Here are some common factorizations worth memorizing:

  • $$(a-b)(a+b)=a^2-b^2$$
  • $$(a \pm b)^2 = a^2 \pm 2ab + b^2$$
  • $$(a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3$$

$\text{Proof }$
$$(a+b)(a^2 - ab + b^2) = a^3 - a^2b + ab^2 + a^2b - ab^2 +b^3 = a^3 + b^3$$
$$\text{Use ($-b$) to replace $b$ in the above, we obtain,}$$
$$(a-b)(a^2 + ab + b^2) = a^3 - b^3$$


  • $$(a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3$$
$\text{Proof }$
$$(a+b)^3 = (a + b) \cdot (a + b)^2 = (a+b)(a^2+2ab+b^2)$$
$$(a+b)^3 = a^3 + 2a^2b+ab^2+a^2b+2ab^2+b^3$$
$$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$
$$\text{Use ($-b$) to replace $b$ in the above, we obtain,}$$
$$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$


  • $$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$$
$\text{Proof }$
$$(a+b+c)^2 = \Big[(a+b)+c\Big]^2 = (a+b)^2 + 2(a+b)(c)+c^2$$
$$(a+b+c)^2 = a^2+2ab+b^2+2ac+2bc+c^2$$
$$(a+b+c)^2 = a^2+b^2+c^2+2ab+2ac+2bc$$

Derived Basic Formulae
  • $a^2+b^2 = (a \pm b)^2 \mp 2ab$
  • $(a+b)^2-(a-b)^2=4ab$
  • $a^3 \pm b^3 = (a+b)^3 \mp 3ab(a+b)$
  • $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-ac-bc)$

Examples
Ex 1.  Evaluate the expression $(2+1)(2^2+1)(2^4+1) ... (2^{2^{10}} + 1) +1.$

Sol.  By using the formula $(a-b)(a+b) = a^2-b^2$ repeatedly, we get 
$$(2+1)(2^2+1)(2^4+1) ... (2^{2^{10}} + 1) +1$$
$$ = (2-1)(2+1)(2^2+1)(2^4+1) ... (2^{2^{10}} + 1) +1$$
$$ = (2^2-1)(2^2+1)(2^4+1) ... (2^{2^{10}} + 1) +1$$
$$ =\text{ ... }= (2^{2^{10}}-1)(2^{2^{10}}+1)+1$$
$$ = ((2^{2^{10}})^{2}-1)+1 = 2^{2 \cdot 2^{10}} = 2^{2^{11}} = \boxed{2^{2048}}$$

Ex 2.  Given $x+\frac{1}{x} = 3$, find the value of $x^3 + \frac{1}{x^3}$.

Sol.  $$x^3 + \frac{1}{x^3} = (x + \frac{1}{x})(x^2 + \frac{1}{x^2} - 1) = 3\Big[(x+\frac{1}{x})^2-3\Big]$$
$$ = 3(3^2-3) = \boxed{18}$$

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Equilateral and Equiangular Polygons

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Incenter/Excenter Lemma

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