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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Basics to get you started

Lets start with some basics that are foundational to some early math competitions. 

Know your Number Line well.

           Natural (Counting) Numbers:
                  1, 2, 3, 4, ...

           Whole Numbers (All Natural Numbers and Zero):
                  0, 1, 2, 3, 4, ...

           Opposites are the same distance from 0, but on opposite sides of zero

           Integers (Whole Numbers and their opposites):
                 ..., -3, -2, -1, 0, 1, 2, 3, ...

           Negative Integers: Integers that are less than 0.

           Positive Integers: Integers that are greater than 0.

Consecutive Numbers: A sequence of numbers that differ only by 1, such as 2, 3, 4, 5.
Consecutive Even Numbers: A sequence of even numbers that differ by 2, such as 2, 4, 6, 8.
Consecutive Odd Numbers: A sequence of odd numbers that differ by 2, such as 1, 3, 5, 7.

Terms to Know 
- Sum: The result that you get when you add two or more numbers.
Ex. 5 is the sum of 2 and 3.

 - Difference: The result that you get when you subtracts numbers.
Ex. 3 is the difference of 5 and 2.

- Product: The result you get when you multiply two numbers.
Ex. 6 is the product of 2 and 3.

- Quotient: The result you get when you divide two numbers.
Ex. 3 is the quotient when you divide 6 and 2.

Multiples, Factors, and Divisibility
The product of any two numbers is a multiple of both those numbers. Furthermore, the two numbers are factors of the product. If a number evenly divides into another number, the first is divisible by the second number. For example, 6 is divisible by 3 because 6 ÷ 3 = 2. 

1. A Prime Number is a counting number that has 2 factors (1 and itself). 
2. A Composite Number is a counting number that has more than 2 
(Note: The number 1 is neither prime or composite because it has only 1 factor, 1.)
3. The Prime Factorization of a number is when the number is written as the product of all primes.
4. The Greatest Common  Factor (GCF) of 2 numbers is the largest factor that evenly divides both numbers, while the Least Common Multiple (LCM) of 2 numbers is the smallest number that is  a multiple of both numbers.

Fractions
1. A common fraction is a fraction where the numerator and the denominator are both whole numbers, but the denominator can't be 0. 
2. A unit fraction is a fraction where the numerator is 1
3. A proper fraction is a fraction where the numerator is smaller than the denominator
4. An improper fraction is a fraction where the numerator is equal to or larger than the denominator.
5. A complex fraction is a fraction that contains a fraction in the numerator or denominator, or both.
6. A simplified fraction is a fraction such that the numerator and denominator share no common factor besides for 1. In other words, the GCF of the numerator and denominator is 1.

Statistics and Probability
The average(arithmetic mean) of a set of numbers is the sum of the numbers divided by the number of numbers in the set. The mode is the number that appears the most. Lastly, the median is the middle number if the set has an odd number of numbers, and the average of the middle two numbers if there are an even amount of numbers.

The probability of an event is a value between 0 and 1 that express how likely an event is to happen, with 0 being it is impossible for the event to happen, and 1 means that it will definitely happen. The probability can often be found by dividing the amount of times and event does occur by the the amount of times the event can possibly occur. For example, the probability of rolling a 1 on a 6 - sided dice is $\frac{1}{6}$ because there is 1 way to role a 1, and 6 different possible outcomes.

Geometry
Angles - The space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet.
Vertex - The point where two lines intersect.
Acute angle - An angle that has a measure of less than $90^{\circ}$.
Right angle - An angle that has a measure of exactly $90^{\circ}$.
Obtuse angle - An angle that has a measure of greater than $90^{\circ}$.
Straight angle - An angle that has a measure of exactly $180^{\circ}$.
Congruent angle - If two angles have the same measure, then they are called congruent.
Congruent line - If two line segments have the same length, then they are congruent.

 Polygons - A polygon is a plane figure that is described by a finite number of straight line segments connected to form a closed figure. (Ex. square, hexagon, octagon.)
Diagonal - A segment that connects two non-adjacent vertices in a polygon.

Area - The amount of unit squares contained in the interior of the region.
Perimeter - The total length of the boundary of a plane figure.
Circumference - The perimeter of a circular region.

You can imagine a polygon as a yard. The amount of grass you have to cut is the area of the lawn, while the amount of fencing needed around the grass is the perimeter.







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Equilateral and Equiangular Polygons

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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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