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<p>In previous research on geometric theorem proving, the primary focus has been on the feasibility of successfully proving specific theorems. Typically, once a theorem was successfully proven, little attention was given to revisiting the process to explore the potential value of the proof results. However, effectively leveraging the geometric knowledge obtained from the proving process is a crucial and underexplored research area. Building upon the complex number identity method—an automated geometric theorem proving approach—and the corresponding complex number identities derived from geometric theorems, this paper introduces an automatic geometric theorem extension algorithm. The algorithm begins by transforming the obtained complex number identity and reinterpreting the geometric significance of each complex number term. It then determines the coordinates of points that satisfy the given geometric conditions by solving equations. Finally, it employs coordinate plotting to generate new geometric propositions. Experimental evaluations on over 300 geometric theorems demonstrate that this automatic extension algorithm not only expands a single geometric theorem into multiple related propositions but also broadens its applicability and reveals underlying connections among different geometric theorems.</p>

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Heuristic strategies for geometry theorem extension based on complex number identity*

  • 期刊论文

  • 发表日期：

    2025年7月29日

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  • Xicheng Peng,
  • Jingzhong Zhang,
  • Mao Chen,
  • Sannyuya Liu

摘要

In previous research on geometric theorem proving, the primary focus has been on the feasibility of successfully proving specific theorems. Typically, once a theorem was successfully proven, little attention was given to revisiting the process to explore the potential value of the proof results. However, effectively leveraging the geometric knowledge obtained from the proving process is a crucial and underexplored research area. Building upon the complex number identity method—an automated geometric theorem proving approach—and the corresponding complex number identities derived from geometric theorems, this paper introduces an automatic geometric theorem extension algorithm. The algorithm begins by transforming the obtained complex number identity and reinterpreting the geometric significance of each complex number term. It then determines the coordinates of points that satisfy the given geometric conditions by solving equations. Finally, it employs coordinate plotting to generate new geometric propositions. Experimental evaluations on over 300 geometric theorems demonstrate that this automatic extension algorithm not only expands a single geometric theorem into multiple related propositions but also broadens its applicability and reveals underlying connections among different geometric theorems.