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Graphing a circle

A circle is easily created using a compass.  A circle is the set of points that are equidistant from a given center. Using the Euclidean distance formula , we can an equation to represent a circle on the Cartesian coordinate system where the center is at $(0,0)$. For radius $r$, the circle would be the set of points that have a Euclidean distance of $r$ from the $(0,0)$. So, we start with the Euclidean Distance formula with:  $r = \sqrt{(x-0)^2 + (y-0)^2}= \sqrt{x^2 + y^2}$ Squaring both sides gives us: $$r^2 = x^2 + y^2$$
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Euclidean Distance

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The Cartesian coordinate system allows us to specify any point using using $(x,y)$ where the center of the coordinates is $(0,0)$  Using this system, we can divide a geographic map into coordinates. It is of course very useful to determine the distance between any two points.  To determine this, we can use the Euclidean Distance formula which is based on the Pythagorean Theorem . The shortest distance between any two points is the size of the hypotenuse which is formed from the triangle made from the two points in question. The distance hence is the square root of the sum of the squares which is equal to $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$
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Pythagorean Theorem

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One of the simplest and most important proofs is the  Pythagorean Theorem .  It provides a valuable engineering insight on how to create a right angle using any material to form the angle and a measuring device.  It is named after the legendary Greek mathematician Pythagoras of Samos who according to legend was the first to provide a proof of this theorem. A right triangle is any triangle where one of its interior angles is 90 degrees. Clearly, there can only be one such angle.  If there were two, then the two opposite lines forming the right angles would be parallel and not be able to form a triangle (this idea is known as the parallel postulate and while it is intuitive, its proof requires some very non-trivial assumptions). When we look at a right triangle, we can see that the line opposite the right angle is the longest side which is called the hypotenuse.  According to Wikipedia , this name is derived from the Greek for "subtending the right angle". ...
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Introduction

Modern mathematics begins with the invention of the mathematical proof.  For further details, check out the Wikipedia article on mathematical proofs . The most well known book of proofs is Euclid's Elements which systematized Greek mathematics and proofs.  This phenomenal work is available online  or pdf form . In this blog, I will present proofs in a manner targeted towards amateurs.  My main purpose is for my own enjoyment but I am very glad if others share my interest in proofs. For those serious about the state of mathematics or professional proofs, I encourage you to read the blog of Terence Tao which is not only authoritative but incredibly well-written. Cheers.
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