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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Circular Permutations

Circular Permutations are type of arrangements where the objects are to be arranged around a circle or in a circular order. Observe that in circular permutations the order around the circle (or the relative positions ) alone need to be taken into consideration and not the actual positions.

For example, suppose $5$ different things are arranged around a circle.

Consider the 5 positions around a circle. $A$, $B$, $C$, $D$, $E$ can be arranged in 5 different positions in 5! ways. Consider one such arrangement say $ABCDE$ in that order around the circle. This arrangement and the $4$ new arrangements $BCDEA$, $CDEAB$, $DEABC$, and  $EABCD$ are not only really different arrangements because the same relative positions around the circle are maintained by the $5$ letters. Therefore, the number of different ways of arranging the  $5$ letters around a circle is $ \cfrac {5!}{5} = 4!$

Also, in the above we are considering the clockwise and anti-clockwise arrangements on the circle to be different from each other, i.e., the arrangement $ABCDE$ clockwise is different from $ABCDE$ anti-clockwise or we may say that we consider arrangement $ABCDE$ different from $EDCBA$. But in some cases we may find that both these arrangements can be obtained without actually changing the positions of the objects involved.

For example, in a beaded bracelet made of five different stones, overturning the bracelet will results in clockwise or anti-clockwise arrangement accordingly. The same cannot be the case in people or any other objects. We now have the general result for $n$ things.

$(i)$ The number of ways of arranging $n$ different things around a circle is $(n-1)!$ or $ \cfrac {1}{2} (n-1)!$ according as whether the counter clockwise and clockwise directions are considered different or same.

$(ii)$ The number of ways of arranging $n$ different things taken $r$ at a time around a circle is $ \cfrac {^nP_{r}}{r}$ if counter clockwise and clockwise directions are considered different and $ \cfrac {^nP_{r}}{2r}$ if they are considered not different.

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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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