Constant direction

We start from a proposal by George A. Tsiamas in Facebook

 We give a proof by using barycentric coordinates. In addition we generalize  the problem for the case in which circles have different radii and finally we consider a homography from line $BC$ to line at infiinity that establishes a relationship between the ratio of the radii of the two circles and the constant direction of the lines joining the midpoints considered in the statement.

Constant direction

Nguyen 046: a generalization leads to a fistful of loci

We start from a proposal by David Nguyen and a generalization gives some loci that satisfy the same property than the centroid:

• Three hyperbolas
• The circumconic with center the symmedian point
• A quartic.

Read here the details: Nguyen046

Envelope of trilinear polars of isogonal conjugates of points on a circle

Given a circle, the envelope of the trilinear polars of the isogonal conjugates of points on the circle is a conic .

The conic, to be a parabola, needs that the given circle goes through the symmedian point $K$ .

Given a circle through $K$ centered at $Q$, call $F'$ the second intersection of $KQ$ and Jerabek hyperbola, and $F''$ the reflection of $F'$ in the midpoint of $K$ and $X_{5505}$. Then the focus $F$ of the parabola lies on line $KF''$ .

Then line joining $F'$ and $X_{5486}$ is parallel to the axis of the parabola .

The point $X_{5505}$ is the Kirikami concurrent circles image of $K$.

In general, let $P$ be a point in the plane of triangle $ABC$ . Let $H_a$ be the orthocenter of triangle $PBC$, and define $H_b$ and $H_c$ cyclically . Let $O_a$ be the circle through the points $A$, $H_b$, $H_c$, and define $O_b$ and $O_c$ cyclically . The circles $O_a$, $O_b$, $O_c$ concur at the Kirikami concurrent circles image of $P$.

The point $X_{5486}$ is the Kirikami Euler image of $K$ .

In general, let $P$ be a point in the plane of triangle $ABC$ . Let $H_a$ be the orthocenter of triangle $PBC$, and define $H_b$ and $H_c$ cyclically. The Euler lines of the triangles $AH_bHc$, $BH_cH_a$, $CH_aH_b$ concur at the Kirikami-Euler image of $P$.

Calculations with Mathematica (pdf version here)

An interpretation of a Nguyen perspector

David Nguyen is an optometrist from Sydney who is very fruitful in discovering concurrences in the also fruitful world of Triangle Geometry. We give an interpretation to one of his recent findings.

An interpretation of a Nguyen perspector

Another relationship between Napoleon cubic and Neuberg cubic

The world of Triangle Geometry is very intrincate. There are many paths that lead to the same place.

In this case a problem from proposed by Benjamin L. Warren at  Euclid 8052 and later expanded by Antreas Hatzipolakis lead to a relationship between these two cubics.

Another relationship between Napoleon cubic and Neuberg cubic

 

La página web de Triángulos Cabri

El Laboratorio Virtual de Triángulos con Cabri fue un sitio web dedicado a la resolución de problemas relacionados con el triángulo.

Estaba dirigida por el profesor Ricardo Barroso Campos y estuvo activa desde 2000 a 2023.

Aunque el sitio ha desaparecido, queda una copia de seguridad en un archivo comprimido. Me tomado, con mucho agrado, la molestia de editar los enlaces que apuntaban a direcciones de la web original a archivos que pueden verse localmente.

Naturalmente habrá errores, algún archivo que falte, etc. Si ves alguno, comunícamelo a 'garciacapitan arroba gmail punto com' para intentar subsanarlo.

Dedicado al profesor Ricardo Barroso Campos y a la gran cantidad de entusiastas que disfrutamos y aprendimos durante todos esos años.

NTC20250219.zip (636 Mb)

Constant angles from the vertices to two cubics

 This is an ancient question that easily can call your attention (Journal de Mathématiques Élémentaires, 1893, p. 254):

We solve it by using barycentric coordinates. Any of the points describe a cubic.

Read the details here. Calculations with barycentric coordinates and Mathematica and comments in Spanish are here.