Construction of regular polygons

There are a few classic problems of ancient mathematics that are easy to state but incredibly difficult to solve. Take for example, the problem of constructing regular polygons and the problem of trisecting an angle by compass and straightedge. It was not until the 18th-19th centuries that mathematicians could finally solve them by employing advanced tools of number theory and algebra. Today, we will look at the regular polygon construction problem.

Regular polygon construction problem. Using compass and straightedge, construct a regular polygon with $n$ sides.

Happy New Year!!!

Pythagorean triples

In geometry, there is a well known theorem, called the Pythagorean Theorem, which says that in a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides. 

Pythagorean Theorem: $BC^2 = AB^2 + AC^2$

That is why we call the equation $$x^2 + y^2 = z^2$$ the Pythagorean equation and its solution $(x,y,z)$ is called a Pythagorean triple. Of course, we only consider integer solutions.

Today, we will solve the Pythagorean equation and show that this equation has an infinite number of solutions.

Wilson's Theorem

Today, we will study Wilson's Theorem - a theorem concerning prime numbers. Wilson's theorem says that if $p$ is a prime number then the number $(p-1)! + 1$ will be divisible by $p$.

Here, the notation $n!$ denotes $$n! = 1 \times 2 \times 3 \times \dots \times n.$$

For example,

• $1! + 1 = 2$ is divisible by $2$
• $2! + 1 = 3$  is divisible by $3$
• $4! + 1 = 25$  is divisible by $5$
• $6! + 1 = 721$  is divisible by $7$

The Fermat's point of a triangle II

In the previous post, we have analyzed the Fermat's problem, the problem of finding a point $M$ for a given triangle $ABC$ such that $MA + MB + MC$ is the minimum.

the Fermat's problem: find $M$ so that $MA + MB + MC$ is minimum

The Fermat's point of a triangle

In previous posts about modulo, we learn about the mathematician Fermat and his famous problem $$x^n + y^n = z^n.$$

Today, we will look at a geometry problem that bears his name. As we already know, Fermat was not a professional mathematician, but was a lawyer. He was doing math probably just for fun and most of his achievements that we know of today originated from his letters to his friends and also from his occasional writings on the margin of books that he read. The most famous is, of course, the problem $x^n + y^n = z^n$ and his note "I have found a beautiful proof but there is not enough space" that he wrote on the margin of the book by Diophantus.

The problem that we investigate today was raised in a letter that Fermat sent to an Italian mathematician, Torricelli. In his letter, Fermat challenged Torricelli to find a point such that the total distance from this point to the three vertices of a triangle is the minimum possible. Well, this problem was not hard for Torricelli. Since Torricelli knew how to find such a point, today some people refer to this point as the Fermat's point, and others refer it as the Torricelli's point of the triangle.

the Fermat's problem: find a point $M$ so that $MA + MB + MC$ is minimum