Rational Angle Bisection and Incenters in Higher Dimensions
Published:Dec 31, 2025 06:14
•1 min read
•ArXiv
Analysis
This paper extends the classic rational angle bisection problem to higher dimensions and explores the rationality of incenters of simplices. It provides characterizations for when angle bisectors and incenters are rational, offering insights into geometric properties over fields. The generalization of the negative Pell's equation is a notable contribution.
Key Takeaways
- •Generalizes the rational angle bisection problem to n-dimensional space.
- •Provides characterizations for rational angle bisectors and incenters.
- •Offers a formula for integral solutions of a generalized negative Pell's equation.
- •Establishes a condition for the rationality of incenters of simplices.
- •Connects the findings to properties of triangles with rational vertices and incenters.
Reference
“The paper provides a necessary and sufficient condition for the incenter of a given n-simplex with k-rational vertices to be k-rational.”