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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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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 ...
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Permutations - When some objects are Not Distinct

Let us consider the following example - In how many different ways can the letters of the word 'POSSESSIVE' be arranged? Observe that there are $10$ letters in the word and all these letters are not distinct. There are $4$ $S's$ and $2E's$ and other letters are distinct. If all the letters were distinct, the number of different arrangements is obviously $^{10}P_{10}$ or $10!$. In this case, since some of the letters are alike the number of arrangements cannot be $10!$  Let us assume that the number of arrangements of the letters of the word 'POSSESSIVE' is $x$. Consider any of these permutations (or arrangements) A typical arrangement can be thought as PSOESVSIES. Keep all except the three $S's$ fixed in their places.  Suppose we replace the four $S's$ in the above by four distinct objects, say $S_{1}$,  $S_{2}$,  $S_{3}$,  $S_{4}.$ If we permute these objects among themselves in all possible ways, it will give rise to $^{4}P_{4}$    or $4!$ arra...
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Circular Permutations

In circular permutations, things are to be arranged in the form of a ring or a circle, e.g. arrangements of people around a circular table. In circular permutation there are no end points, i.e. there are no beginning or ending positions. So, the number of circular permutations of n objects are $$\frac{n!}{n}=(n-1)!$$ Thus in a circular permutation, one thing is kept fixed and the remaining  $(n-1)$ things are arranged in $(n-1)!$ ways. If the clockwise and counter clockwise orders are not distinguishable, then the number of ways = $\frac{1}{2}(n-1)!$ Let us look at a few examples.  Ex:   In how many ways can $6$ boys be seated at a circular table? Sol:   We keep one boy in a fixed position and to find the number of permutations to arrange the remaining 5 boys, we simply take $5!$, which gets us to our answer $\boxed {120} $.  Ex:  In how many ways can $6$ boys be arranged at a round table so that $2$ particular boys can be seat...
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Permutations

Each of the different arrangements that can be made out of a given number of things by taking some or all of them at a time is called a permutation. Thus the permutation of the three letters $a, b, c$ taken two at a time are: $$ab, ba, bc, cb, ac, ca$$ Therefore the number of permutations of three different things taken two a time is $_{3}P_{2}$ or $P(3, 2) = 6$. We can generalize this to say $_{n}P_{r} =$ Total number of permutations of $n$ distinct things taken $r$ at a time. $$_{n}P_{r} = \frac{n!}{(n-r)!}$$ Below are some extensions of the above concept. Permutations of $n$ different things taken all at a time $$ = _{n}P_{n} = n!$$ Ex . Find the number of ways the letters of the word RAINBOW can be rearranged. Sol . Every letter in the word RAINBOW is different, so to get the number of permutations, we would simply use $n!$. $$n = 7 \Rightarrow n! = \boxed{5040}$$ Permutations of $n$ different things taken $r$ at a time, when $k$ things never occurs $$ = _{n-k}P_{...
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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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More Coordinate Geometry

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We have looked at some basics of coordinate geometry. Today, let's look at some intermediate applications of this topic. 1. Shoelace Theorem The Shoelace Theorem is an algorithm used to find the area of a polygon in the coordinate plane when the coordinates are known. If the coordinates are $(x_1, y_1), (x_2, y_2) ... (x_n, y_n), $ $\text{Area}=\frac{1}{2}\Big[(x_1y_2 - x_2y_1)+(x_2y_3 - x_3y_2) + ... + (x_ny_1 - x_1y_n)\Big].$ 2. Centroid The centroid is the point where the three medians intersect. It is also sometimes called the center of gravity for the triangle. Note: A median of a triangle is the line segment joining the vertex to the midpoint of the opposite side. If $(x_1,y_1), (x_2, y_2),$ and $(x_3,y_3)$ are the vertices of a triangle, then the coordinates of its centroid are $$\Bigg(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3} \Bigg).$$ 3. Incenter The incenter is the point where the three angle bisectors intersect. Note: An angle bisector of a triangle...
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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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