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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Modular Arithmetic


1. Let $n$ be a fixed positive number. Two integers $a$ and $b$ are said to be congruent modulo $n$, symbolized by $a\equiv b \pmod{n}$ if $n$ divides the difference $a-b$., i.e., provided that $a-b = kn$ for some integer $k$.
    • For ex.  $23\equiv 3 \pmod{5}$, $19\equiv 3 \pmod{4}$,  $12\equiv 5 \pmod{7}$,  $5\equiv 5 \pmod{3}$,
2. For arbitrary integers $a$ and $b$, $a\equiv b \pmod{n}$ iff $a$ and $b$ leave the same non negative remainder when divided by $n$


3. Let $n>1$ be fixed and $a, b, c, d$ be arbitrary integers. Then the following properties hold:

  • $a\equiv a \pmod{n}$
         Ex., $7\equiv 7 \pmod{3}$
         $7-7 = 0$ is a multiple of $3$. Hence $3$ divides $0$

  • If $a\equiv b \pmod{n}$, then $b\equiv a \pmod{n}$
         Ex., $17\equiv 3 \pmod{7}$
         This congruence is true, since, $17-3 = 14$ is a multiple of $7$.
         Also, $3\equiv 17 \pmod{7}$ is true, since $3-17 = -14$ is divisible by $7$, the quotient is $-2$

  • If $a\equiv b \pmod{n}$ and $b\equiv c \pmod{n}$ then $a\equiv c \pmod{n}$
         Ex., Consider the congruences $19\equiv 11 \pmod{4}$ and $11\equiv 3 \pmod{4}$,
         the congruence is $19\equiv 3 \pmod{4}$ is also true.

  • If $a\equiv b \pmod{n}$ and $c\equiv d \pmod{n}$, then $a+c\equiv b+d \pmod{n}$ and $ac\equiv bd \pmod{n}$
         Ex., Consider the following congruences
         $7\equiv 2 \pmod{5}$    .......(1)
         $13\equiv 3 \pmod{5}$    .......(2)
         Adding eq. $(1)$ and $(2)$ , we get
         $7+13\equiv 2+3 \pmod{5}$
         $20\equiv 5 \pmod{5}$, which is true.

         Multiplying eq. $(1)$ and $(2)$, we get
         $7*13\equiv 2*3 \pmod{5}$
         $161\equiv 6 \pmod{5}$ , which is true.

  • If $a\equiv b \pmod{n}$ , then $a+c\equiv b+c \pmod{n}$ and $ac\equiv bc \pmod{n}$
         Ex., Consider the congruence $17\equiv 1 \pmod{8}$
         Adding 3 to both sides, we get
         $17+3\equiv 1+3 \pmod{8}$
         $20\equiv 4 \pmod{8}$
         which is true , since $20 - 4 = 16$ is a multiple of $8$.
         Now, multiplying 3 to both sides of $17\equiv 1 \pmod{8}$, we get
         $17*3\equiv 1*3 \pmod{8}$
         $51\equiv 3 \pmod{8}$
         which is true, since $51-3 = 48$ is a multiple of $8$.

  • If $a\equiv b \pmod{n}$, then $a^k\equiv b^k \pmod{n}$, for any positive integer $k$.
         Ex., Consider the congruence $5\equiv 2 \pmod{3}$
         On raising both sides with any positive integer, let it be $2$, we get
         $5^2 \equiv 2^2 \pmod{3}$
         $25 \equiv 4 \pmod{3}$, which is true.
         Similarly, $5^4 \equiv 2^4 \pmod{3}$
         $625 \equiv 16 \pmod{3}$
         which is true, since, $625 - 16 = 609$ is a multiple of $3$.

  • If $ca \equiv cb \pmod{n}$, then $a \equiv b \pmod{\frac{n}{d}}$, where $d=gcd(c,n)$
          Ex., Consider the congruence $64 \equiv 28 \pmod{6}$
          Letting $c=4$ and dividing, we get
          $16 \equiv 7 \pmod 3$ because $64 \div 4 = 16$, $28 \div 4 = 7$, and $gcd(4, 6) = 2$         









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Equilateral and Equiangular Polygons

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