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

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 is the line segment that equally divides one of the vertex angles of the triangle.

If $A(x_1, y_1), B(x_2, y_2)$, and $C(x_3, y_3)$ are the vertices of the triangle $ABC$ such that $BC = a, CA = b,$ and $AB = c$, then the coordinates of its incenter are
$$\Bigg(\frac{ax_1+bx_2+cx_3}{a+b+c}, \frac{ay_1+by_2+cy_3}{a+b+c} \Bigg).$$

4. Point of Intersection of Two Lines

To find the coordinates of the point of intersection of two non-parallel lines, we solve the given equations simultaneously and the values of $x$ and $y$ obtained determine the coordinates of the point of intersection.

5. Distance between Two Parallel Lines

If two lines are parallel, then they have the same distance between them throughout. Therefore, to find the distance between any two parallel lines, we find the coordinates of any point on either of the given lines, preferably letting $x=0$ or $y=0$. Then the perpendicular distance from this point to the other line is the required distance between distance.
     Alternatively: Let the two parallel lines be $ax+by+c_1 = 0$ and $ax+by+c_2=0$, then the perpendicular distance between the lines is $$\Bigg | \frac{c_2-c_1}{\sqrt{a^2+b^2}} \Bigg |.$$

6. Classifying Polygons 

  • If the vertices of a triangle have integral coordinates, then the triangle can not be equilateral. 
  • In order to prove that a given figure is a square, parallelogram, rectangle etc, we will need to prove the below properties.

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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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