Undergrad Integer Cevians - Equilateral Triangles

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SUMMARY

The discussion focuses on the theory of integer cevians in equilateral triangles, specifically regarding the number of integer cevians that divide the triangle's sides into integer parts. A notable example is provided where an equilateral triangle with a side length of 8 has a cevian of 7 that partitions one side into segments of 3 and 5. References include the OEIS sequence A089025 and various academic papers that explore integer triangles with integer medians and rational altitudes. Additionally, a GitHub repository offers a comprehensive collection of equilateral triangles and their cevians, including data structures in Racket, JavaScript, and Java.

PREREQUISITES
  • Understanding of integer cevians in geometry
  • Familiarity with equilateral triangles and their properties
  • Basic knowledge of programming languages such as Racket, JavaScript, and Java
  • Access to mathematical resources like OEIS and academic papers
NEXT STEPS
  • Research the OEIS sequence A089025 for further examples of integer cevians
  • Explore the paper on integer triangles with integer medians for deeper insights
  • Examine the GitHub repository for practical applications and data structures related to cevians
  • Learn about the mathematical properties of rational altitudes and angle bisectors in triangles
USEFUL FOR

Mathematicians, geometry enthusiasts, and programmers interested in computational geometry and the properties of integer cevians in equilateral triangles.

danieldf
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Is anybody familiar with any theory of integer cevians on equilateral triangles?

More specificaly, I was trying to find something about the number of integer cevians that divide the side in integer parts. Like, the eq triangle of side 8 have cevian 7 dividing one side into 3+5.

Only reference I found is here https://oeis.org/A089025
Where it examples the triangle of side 280 having cevians "247 partitioning an edge into 93+187, as well as cevian 271 that sections the edge into 19+261."
But nothing else online on how to get to these partitions or how many one could find for a specific triangle.
 
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Here is a repo with these equilateral triangles and their Cevians with all such instances going up to 7^4:

https://github.com/jprus/eisenstein-triples

It includes the complementary orthogonal Cevian multiplied by sqrt(3). In both cases you have the integer length of the sides of the equilateral triangle as well as the lengths of the partitioned sides all matched to a particular cevian of form "6i+1 prime" (as well as composite numbers with all factors of this form). The information is presented as an array of objects/structs in Racket (ie. modern DrScheme, a language popular in academia and functional programming), JavaScript, and Java so you can continue to experiment with these.
 
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