## Some results related to the Feuerbach Point

Given a triangle the circle that goes through the midpoints of each side, is called the Feuerbach circle. It has very surprising properties:

- It also goes through the feet of the heights, points
- If denotes the orthocenter of the triangle, then the Feuerbach circle also goes through the midpoints of the segments For this reason, the Feuerbach circle is also called the
**nine-point circle.** - The center of the Feuerbach circle is the midpoint between the orthocenter and circumcenter of the triangle.
- The area of the circumcircle is precisely four times the area of the Feuerbach circle.

Most of these results are easily shown with `sympy` without the need to resort to Gröbner bases or Ritt-Wu techniques. As usual, we realize that the properties are independent of rotation, translation or dilation, and so we may assume that the vertices of the triangle are and for some positive parameters To prove the last statement, for instance we may issue the following:

>>> import sympy >>> from sympy import * >>> A=Point(0,0) >>> B=Point(1,0) >>> r,s=var('r,s') >>> C=Point(r,s) >>> D=Segment(A,B).midpoint >>> E=Segment(B,C).midpoint >>> F=Segment(A,C).midpoint >>> simplify(Triangle(A,B,C).circumcircle.area/Triangle(D,E,F).circumcircle.area) 4

But probably the most amazing property of the nine-point circle, is the fact that it is tangent to the incircle of the triangle. With exception of the case of equilateral triangles, both circles intersect only at one point: the so-called **Feuerbach point**.

## Bertrand Paradox

Classically, we define the probability of an event as the ratio of the favorable cases, over the number of all possible cases. Of course, these possible cases need to be all equally likely. This works great for discrete settings, like dice rolls, card games, etc. But when facing non-discrete cases, this definition needs to be revised, as the following example shows:

Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?

First example Second example Third example

## Mechanical Geometry Theorem Proving

In 1977, Professor Wen-Tsun Wu succeeded in developing a method of mechanical geometry theorem proving. This method has been applied to prove or even discover hundreds of non-trivial difficult theorems in elementary and differential geometries on a computer in an almost trivial manner. Usign Ritt’s differential algebra, Wu established a method for solving algebraic and differential equations by transforming an equation system in the general form to equation systems in triangular form. This is the Ritt-Wu decomposition algorithm, that later on was shown to be equivalent to perform a series of operations on ideals, very easily carried out by means of Gröbner basis manipulation.

I wrote a script in `MAPLE`

to perform evaluations of the validity of some simple theorems in Euclidean Geometry, and wrote a small paper (in Spanish) on one of my findings, that was published in Bol. Asoc. Prof. Puig Adams, in October’99: “Sobre demostración automática de un problema geométrico“.

The example I cover in that short article can be seen below.

Given: Circles , that intersect each other in points and , and given points , in circle , consider line through and , and line through and . The intersections of line with circle are and . The intersections of line with circle are and . Consider the segments (connecting with ) and (connecting with ).

To prove: Segments and are parallel.