Due to the properties of the Frégier point, Frégier’s theorem provides a practical means of constructing with straightedge and square the tangent to a conic at any point on the respective conic. Constructing a Frégier point is very easy and the procedure is the same for any type of conic section (parabola, hyperbola, ellipse or circle). Right… Continue reading Frégier’s Theorem and Frégier Points

# Category: geometry

## A Very Easy Method to Inscribe a Regular Pentagon in a Circle

The classical way of inscribing a regular pentagon in a circle is discussed in Book IV, Proposition 11 of Euclid’s Elements. The method of Euclid is a bit complicated since it requires first to draw a golden triangle (a 72-72-36 triangle) and then to inscribe the triangle in the given circle. In this post I… Continue reading A Very Easy Method to Inscribe a Regular Pentagon in a Circle

## Lill’s Method and Geometric Solutions to Quadratic Equations With Complex Roots

I already wrote a paper called “Lill’s Method and Graphical Solutions to Quadratic Equations” that shows how to solve quadratic equations using Lill’s method. In this post I want to show an alternative way of solving quadratic equations that have complex or imaginary roots. This method was briefly mentioned in the paper “Geometric Solution of… Continue reading Lill’s Method and Geometric Solutions to Quadratic Equations With Complex Roots

## The Hexagram/Six Pointed Star/Star of David and the Golden Ratio

Many people that are familiar with the golden ratio ( Φ = 0.5(1+ 50.5) which is about 1.61803), know that it appears very often in the proportions of the pentagon and the pentagram. In this post I want to present a diagram that connects the golden ratio with the regular hexagram or the six pointed… Continue reading The Hexagram/Six Pointed Star/Star of David and the Golden Ratio