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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Polynomials: Key Vocabulary

Let us look at some key terms that pertain to Polynomials before we deep dive into the study of Polynomials. 

 Constant: A number having a fixed numerical value
Example: $3, \cfrac{4}{5}, 4.2, 6.\overline{3}$

 Variable: A number which can take various numerical values
Example:  $ x,  y,  z$

 Algebraic Expression: A combination of constants and variables connected by arithmetic operators
Example: $2x^2 + 7, 5x^3 + 4xy + 2xy^2 + 7,$ etc  

 Terms: Several parts of an algebraic expression separated  by $+$ or $-$ signs are called the terms of the expression. 
Example: In the expression $9x + 7y + 5$, $9x$, $7y$, and $5$ are terms.  

 Coefficient of a Term: In the term $8x^2$, $8$ is the numerical coefficient of $x^2$ and $x^2$ is said to be the literal coefficient of 8. 

Like Terms: Terms having the same literal coefficients are called Like Terms. 
Example: $8xy$, $9xy$, and $10xy$ are Like Terms and $9x^2, -10x^2,$ and $5x^2$ are Like Terms. 

 Unlike Terms: Terms having different literal coefficients are called Unlike Terms. 
Example: $8xy$, $9x^2$, and $10xy^2$ are Unlike Terms. 

Polynomial: An algebraic expression in which the variables involved have only non-negative integral powers. 
Example: $4x^2 + 2x + 1,$ $5x^3 + 16x^2 + 9$

Note - The expression $3x^3 + 4x + \cfrac{5}{x} + 6x^ \frac {3}{2}$ is not a polynomial, since it has powers of $x$ which are negative and fractions. 
      A polynomial that contains only one variable is known as a polynomial in that variable. 

Degree of a Polynomial in one variable: The highest power of the variable in a polynomial of one variable is called the degree of the polynomial. 
Example: $(i)$  $ 2x^3 + 3x^2 + x + 8 $ is a polynomial of degree $3$. 
               $(ii)$  $ 8x^5 + 6x^3 + 7x $ is a polynomial of degree $5$. 


Types of Polynomials with Respect to Degree

$(1)$ Linear polynomial: A polynomial of degree one is called a linear polynomial. 
        Example: $ 2x + 5$,   $6y - 9$,  $7z + 8,$ etc. 

$(2)$ Quadratic polynomial: A polynomial of degree two is called a quadratic polynomial. 
        Example: $ 6x^2 + 5x + 4$,   $8y + 7y^2 + 9$, etc. 

$(3)$ Cubic polynomial: A polynomial of degree three is called a cubic polynomial. 
        Example: $ 7x^3 + 8x^2 + 5x + 3$,   $8z^3 + 9z + 3$, etc. 


Types of Polynomials with Respect to Number of Terms

$(1)$ Monomial: An expression containing only one term. 
        Example: $ 6x$,   $5x^3$,  $7xyz^3,$ etc. 

$(2)$ Binomial: An expression containing two terms. 
        Example: $ 6x^2 + 5y$,   $2xy + 5$, etc. 

$(3)$ Trinomial: An expression containing three terms. 
        Example: $ 7x + 2y - 4z $,   $4z^3 + 5xy + 9$, etc.

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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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Incenter/Excenter Lemma

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Given $\triangle ABC$ with incenter $I$, extend $AI$ to meet the circumcircle of $\triangle ABC$ at $L$. Then, reflect $I$ over $L$, and name this point $I_a$. Then:  $(i)$ $BICI_a$ is a cyclic quadrilateral with point $L$ as the center of the circumscribed circle.  $(ii)$ $BI_a$ and $CI_a$ bisect the exterior angles of $\angle B$ and $\angle C$,  respectively.  Proof $(i)$ Notice that the the question is equivalent to proving the distance from $L$ to points $B, I, C, I_a$ are equal. In other words, we need to show that  $$LB = LI = LC = LI_a$$ Firstly, it is obvious that $LI = LI_a$ because by definition, $I_a$ is the reflection of $I$ over $L$, and reflections preserve the length of the segment.  Now, we are left with proving that $LB=LI$, because similar calculations will provide $LI=LC$.  We see that most of our given information involves angles, so we focus on proving $\angle LBI = \angle LIB$. Firstly,  $$\angle LBC = \angl...
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Irrational Numbers

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