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Theorem Discovery Amongst Cyclic Polygons

Philip Todd

TL;DR

The work tackles automatic discovery of geometric theorems in cyclic $2n$-gons by converting angle-bisection constraints into linear relations among inter-side angles, then generalizing from parallel-conditions to fixed-angle relations via a winding-number framework. It develops a diagrammatic, algorithmic workflow that yields a general theorem: $\sum_{i=1}^n (-1)^{b_i}\delta_{a_i b_i}=W\pi$ for appropriate side-pairings, and demonstrates this with a matrix/triangulation approach. An automated problem generator and a library of examples produce human-readable proofs and ready-made problem collections, illustrating the method's educational and AI-compatibility potential. The results offer a scalable route to generate and solve angle-based geometry problems while highlighting the role of diagram geometry and directed vs. undirected angle conventions in automated reasoning.

Abstract

We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides which is constant. We present a formula for the linear combination, which provides a theorem statement in terms of those angles. We describe a program which uses this result to generate new geometry proof problems and their solutions.

Theorem Discovery Amongst Cyclic Polygons

TL;DR

The work tackles automatic discovery of geometric theorems in cyclic -gons by converting angle-bisection constraints into linear relations among inter-side angles, then generalizing from parallel-conditions to fixed-angle relations via a winding-number framework. It develops a diagrammatic, algorithmic workflow that yields a general theorem: for appropriate side-pairings, and demonstrates this with a matrix/triangulation approach. An automated problem generator and a library of examples produce human-readable proofs and ready-made problem collections, illustrating the method's educational and AI-compatibility potential. The results offer a scalable route to generate and solve angle-based geometry problems while highlighting the role of diagram geometry and directed vs. undirected angle conventions in automated reasoning.

Abstract

We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides which is constant. We present a formula for the linear combination, which provides a theorem statement in terms of those angles. We describe a program which uses this result to generate new geometry proof problems and their solutions.
Paper Structure (11 sections, 2 theorems, 26 equations, 8 figures)

This paper contains 11 sections, 2 theorems, 26 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Cyclic Polygons and Angle Bisectors
  3. Cyclic Quadrilateral
  4. General Theorem
  5. Automated Discovery of Cyclic Polygon Theorems
  6. Choosing Side Pairs
  7. Diagram Creation
  8. Geometric Angle Conclusion
  9. Example
  10. An Automated Problem Generator
  11. Conclusion

Key Result

Theorem 2.1

Given a cyclic 2n-gon with winding number $W$ about the circumcircle's center, and $n$ ordered pairs $(a_1,b_1) \ldots (a_n,b_n)$ such that $\{ a_i \} \cup \{b_i\} = \{ 1 \ldots 2n \}$, $b_i>a_i$ and $b_i-a_i$ is odd for each $i$

Figures (8)

  • Figure 1: $\theta_0 \ldots \theta_6$ are angles of position vectors. $\theta_6=\theta_1+2\pi$. $\phi_i=\frac{1}{2}(\theta_{i-1}+\theta_i)$
  • Figure 2: A cyclic quadrilateral with winding number 1.
  • Figure 3: Graphs representing possible conforming sets of edge pairs for $n=3, 4$. Vertices on these graphs correspond to polygon sides. Edges indicate the side pairs whose angles will be specified
  • Figure 4: Graph for the set of side pairings $\{(1,2),(3,10),(4,7),(5,8),(6,9)\}$
  • Figure 5: Cyclic decagon with angles marked for side pairings $\{(1,2),(3,10),(4,7),(5,8),(6,9)\}$
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • proof
  • Theorem 5.1
  • proof