Posts

Showing posts from August, 2019

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...
Post a Comment
Read more

Coordinate Geometry

Image
Coordinate Geometry is the study of geometry using algebraic tools like equations, relations, and operations. Before we start our deep dive, let's go over some basics of coordinate geometry. A coordinate grid consists of 2 perpendicular lines, which are called axes , and they are labeled like a number line would be. The horizontal axis is known as the x-axis , while the vertical axis is called the y-axis . The point of intersection of the two axes is known as the origin . The coordinate of the origin will always be $(0, 0)$. The numbers along the axes are used to locate points in the plane. The point at which a graph intersects the x-axis is known as the x - intercept  and the point of intersection with the y-axis is known as the y - intercept . The coordinates  also known as an Ordered Pair, is a set of values that expresses the distance a point lies from the origin. It is written in the format $(x, y)$ where $x$ is the value on the x-axis and the $y$ is the value on the ...
Post a Comment
Read more

Circle Theorems

Image
Now that you know the fundamentals of circles, let's take a look at some important circle theorems.  Inscribed angle is half the measure of the intercepted arc. An inscribed angle is the angle formed in the interior of a circle when two secant lines intersect on the circle.  The above theorem leads to an important result that comes handy in solving many geometry problems. The angle at the circumference subtended by a diameter is a right angle.  Simply put, the angle in a semicircle is a right angle.    All inscribed angles that intercept the same arc have the same measure.  The angle at the center is twice the angle at the circumference subtended by the same arc. The measure of the angle formed by the intersection of two chords is equal to the average of it's two corresponding arcs. Simply put, m$\angle \alpha = \frac{\overset{\LARGE{\frown}}{NP} + \overset{\LARGE{\frown}}{QO}}{2}$ Similar to the ...
Post a Comment
Read more

Circle Basics

Image
The past 2 weeks that I have been away, were quite the adventure in the Grand Teton and the Yellowstone National Parks. As a family we have always loved spending time outdoors and this was the perfect trip right before the onset of my sophomore year. Getting back to life always seems so hard after a trip like that. Before I digress too much, lets get back to work. Today I would like to review the fundamentals of Circles. Let's first understand the different components of a Circle. Here are the key terms you should know. Radius - Line Segment from the center of the Circle to a point on the circumference. Diameter - Line Segment from one point on the circumference to another point on the circumference that passes through the center of the Circle. Needless to say, the length of the diameter is twice that of the radius. Area - The set of all points contained inside the circumference. Can be found using the formula $ \pi r^2$ where $r$ is the radius. ...
Post a Comment
Read more

Popular posts from this blog

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...
Read more

Irrational Numbers

A number which cannot be written in the form $ \cfrac {p}{q}$  where $p$ and $q$ are integers and $ q $  $ \neq 0$. Example: $ \sqrt {2}$, $ \sqrt {3}$, $ \sqrt {6}$, $ \sqrt {7}$, $ \sqrt {8}$, $ \sqrt {10}$ Theorem: If $p$ divides $x^3$, then $p$ divides $x$, where $x$ is a positive integer and $p$ is a prime number.  Proof :  Let $x = p_{1} p_{2}...p_{n}$ where $p_{1}, p_{2}, p_{3}.....p_{n}$  are primes, not necessarily distinct.  $ \rightarrow x^3 = p_{1}^3 p_{2}^3.....p_{n}^3$ Given that $p$ divides $x^3$ By fundamental theorem, $p$ is one of the primes of $x^3$.  By the uniqueness of fundamental theorem, the distinct primes of $x^3$ are same as the distinct primes of $x$.  $ \rightarrow p$ divides $x$ Similarly if $p$ divides $x^2$, then $p$ divides $x$, where $p$ is a prime number and $x$ is a positive integer. Example: Prove that  $ \sqrt {2}$ is irrational.  Solution : Let us assume that  $ \sqrt ...
Read more

Incenter/Excenter Lemma

Image
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...
Read more