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Albime Triangles: Acute or Obtuse

Kooij, J. F. (2015) Albime Triangles: Acute or Obtuse. Bachelor's Thesis, Mathematics.

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Abstract

A triangle ABC is called albime if a bisector in A, the median in B and the altitude in C are concurrent. We scale these triangles such that the points of the elliptic curve y^2=x^3-4x+4 that have an x-coordinate between zero and two correspond exactly to an albime triangle with internal bisector, and the point that have an x-coordinate between minus two and zero to an albime triangle with external bisector. The rational points on this curve form a group that can be generated by one element, namely P=(2,2). This group of rational points can be mapped onto the unit circle using some map psi. Then there exists the element psi(P) that generates the unit circle. We can find how many of the rational points on this circle correspond to a rational albime triangle. We find that 36.12% of the points coincide with an albime triangle with inner bisector and 19.40% of them generate one with an external bisector. For the albime triangles with internal bisector we prove that 22.48% of rational points on the elliptic curve coincide with an acute albime triangle and 13.67% with an obtuse albime triangle.

Item Type: Thesis (Bachelor's Thesis)
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 08:04
Last Modified: 15 Feb 2018 08:04
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/12761

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