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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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Circumradius of a Triangle

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In $\bigtriangleup ABC$, bisect the two sides $BC$ and $CA$ in $D$ and $E$ respectively and draw $DO$ and $EO$ perpendicular to $BC$ and $CA$.  Thus, $O$ is the center of the circumcircle. Join $OB$ and $OC$.  The two triangles $BOD$ and $COD$ are equal in all respect, so  $ \angle BOD = \angle COD$ Therefore, $ \angle BOD = \cfrac{1}{2} \angle BOC =  \angle BAC = \angle A$ Also, $BD = BO \sin \angle BOD$ Therefore, $\cfrac{a}{2} = R \sin \angle A$ or $R =  \cfrac{a}{2 \sin \angle A}$ Now, we know that sin $A = \cfrac{2}{bc}$ $\sqrt{s(s-a)(s-b)(s-c)} = \cfrac {2S}{bc}$ where $S$ is the area of the triangle.   Therefore, substituting the above values of sin $\angle A$ in eq $(1)$, we get $R = \cfrac {abc}{4S}$ giving the radius of the circumcircle in terms of the sides. 
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Inradius of a Triangle

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In $\triangle ABC$, let the name the incenter as point $I$ and the feet of radii as $D, E,$ and $F$ as shown below. Since $I$ is the incenter, $ID = IE = IF = r$ Now, using the $A = \frac{bh}{2}$ formula on $\triangle ICB, \triangle IAC, \text{and } \triangle IBA$, we get   $[ICB] = \frac{1}{2}ID \cdot BC = \frac{ra}{2}$ $[IAC] = \frac{1}{2}IE \cdot AC = \frac{rb}{2}$ $[IBA] =  \frac{1}{2}IE \cdot AB = \frac{rc}{2}$ Adding all three of our equations, we have  $$\frac{ra}{2} + \frac{rb}{2} + \frac{rc}{2} = [ABC]$$ Letting $s = \text{semiperimeter} = \frac{a+b+c}{2}$, we get $$[ABC] = r \cdot s$$
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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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