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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More Numbers

Key Facts


1. Numbers of the form $\frac{p}{q}, q \neq 0$, where $p$ and $q$ are integers and those that can be expressed in the form of terminating or repeating decimals are called rational numbers.
            Ex. $\frac{7}{32} = 0.21875, \frac{8}{15} = 0.5\bar{3}$ are rational numbers.

2. Properties of operations of rational numbers
    For any rational numbers $a, b, c,$

    (i) Rational numbers are closed under addition, multiplication, and subtraction.
         $i.e., (a+b), (a-b),$ and $(a \cdot b)$ are all rational

    (ii) Rational numbers follow the commutative law of addition and multiplication.
         $i.e., a + b = b + a$ and $a \cdot b = b \cdot a.$

    (iii) Rational numbers follow the associative law of addition and multiplication,
         $i.e., (a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = a \cdot (b \cdot c).$

    (iv) Additive Identity: $0$ is the additive identity for rational numbers because $a + 0 = 0 + a = 0.$

    (v) Multiplicative Identity: $1$ is the multiplicative identity for rational numbers as $a \cdot 1 = 1 \cdot a = a.$

    (vi) Additive Inverse: For every rational number $a$, there is a rational number $–a$ such that $a + (–a) = 0.$

    (vii) Multiplicative Inverse: For every rational number $a$ except $0$, there is a rational number $\frac{1}{a}$ such that $a \cdot \frac{1}{a} = 1$.

    (viii) Distributive Property: Multiplication distributes over addition in rational numbers,
         $i.e., a (b + c) = a × b + a × c.$

3. Numbers which when converted into decimals are expressible neither as terminating nor as repeating decimals are called irrational numbers.
          Ex. $\sqrt{2} = 1.41421...$ is an irrational number.
An irrational number cannot be expressed in the form $\frac{p}{q}, (q \neq 0)$.

4. The set of all rational and irrational numbers is called real numbers which is denoted by $\mathbb{R}$.

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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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Irrational Numbers

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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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