Equation for the sidelengths of a triangle in terms of its three angle bisectors

We give a 20th degree polynomial one of whose roots is one of the sidelengths of the triangle.

Happy New Year!

 threebisectors.pdf

Euclid 3639 - Some loci of circumcenters

We study a problem proposed by Antreas Hatzipolakis at Euclid 3639 

PDF: Euclid 3639

A circle with huge equation

 Antreas Hatzipolakis proposes the problem Euclid 3637 as follows:

Let ABC be a triangle, L the Euler line, P a point on L and A'B'C' the pedal triangle of P
Denote
A", B", C" = the orthogonal projections of A, B, C, on L, resp.
(Oa), (Ob), (Oc) = the circles with diameters A'A", B'B", C'C", resp.
The radical center of (Oa), (Ob), (Oc) lies on the NPC.
Which point is it in terms of P?

Here is the Mathematica code and its output to the solution of this problem and its generalization for any line $px + qy + rz =0$.

Solution to Euclid Problem 3637

A figure for Problem Euclid 3630

A figure for Problem 3630

Some problems on degenerate triangles

We study by transformations some loci related to the centroid of  degenerate triangles.

Some problems on degenerate triangles

My solution to Euclid 2066

 This is my solution to Problem Euclid 2066 by Antreas Hatzipolakis. The solution contains a conic and a cubic and several new triangle centers.

Solution

The conic:

The cubic:

Revista Escolar de la Olimpiada Iberoamericana de Matemáticas

Pongo aquí enlaces a mis artículos publicados en la desaparecida Revista Escolar de la Olimpiada Iberoamericana de Matemáticas dirigida durante años por el profesor Francisco Bellot Rosado:

• Inversión en Olimpiadas (Aplicación de la inversión a la resolución de problemas)
• Lectura atenta del Court:
  • Catorce Problemas del Court
  • El triángulo órtico
  • La recta de Simson en el Court
• Ángulos e inversión
• Un problema del Sidler
• Algunas homografías en el triángulo
• Máximos sin derivadas

(22/07/2022) ¡Novedad! Todos los archivos de la la Revista pueden descargarse aquí: REOIM.ZIP (100 Mb).

Coordenadas baricéntricas

Barycentric coordinates allow you easily make calculations in triangle geometry. A document on barycentric coordinates (Spanish). cb.pdf This document come from a reading of the book Introduction to the Geometry of the Triangle by Paul Yiu. We solve some of the proposed exercises and we include some examples of other sources. Some Mathematica routines to work on barycentric coordinates.  baricentricas.nb Installation: Copy the notebook into a directory where Mathematica can find it, such as the AddOns\ExtraPackages. In a standard Windows installation this can be the directory C:\Archivos de programa\Wolfram Research\Mathematica\5.2\AddOns\ExtraPackages Open and execute the notebook on that location. A file baricentricas.m will be generated Now you can call the routines in baricentricas.nb from any other notebook by entering the command <<Baricentricas`;