On triangles with rational altitudes, angle bisectors or medians

Elliptic Curves and Triangles with Three Rational Medians

In his paper Triangles with three rational medians, about the characterization of all rational-sided triangles with three rational medians, Buchholz proves that each such triangle corresponds to a point on a oneparameter family of elliptic curves whose rank is at least 2. We prove that in fact the exact rank of the family in Buchholz paper is 3. We also exhibit a subfamily whose rank is at leas...

Some Geometric Inequalities Involving Angle Bisectors and Medians of a Triangle

In the present paper some geometric inequalities concerning the angle bisectors and the medians of a triangle are established. Mathematics subject classification (2010): Primary 26D15.

Subdivision Using Angle Bisectors Is Dense in the Space of Triangles

Starting with any nondegenerate triangle, we can use an interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show that starting with any arbitrary triangle, the resulting set of triangles formed by this process contains triangles arbitrarily close (up to similarity) to any given tria...

Rational Triangles with Equal Area

We consider the set of triangles in the plane with rational sides and a given area A. We show there are infinitely many such triangles for each possible area A. We also show that infinitely many such triangles may be constructed from a given one, all sharing a side of the original triangle, unless the original is equilateral. There are three families of triangles (including the isosceles ones) ...

On a question of Gowers concerning isosceles right-angle triangles