作者投稿和查稿 主编审稿 专家审稿 编委审稿 远程编辑

计算机工程 ›› 2025, Vol. 51 ›› Issue (7): 78-89. doi: 10.19678/j.issn.1000-3428.0068959

• 人工智能与模式识别 • 上一篇    下一篇

基于向量恒等式的几何证明题自动生成方法

李雷1,2, 陈矛2,3,*(), 彭翕成2,3   

  1. 1. 西北师范大学教育技术学院,甘肃 兰州 730070
    2. 华中师范大学教育大数据应用技术国家工程研究中心,湖北 武汉 430079
    3. 华中师范大学人工智能教育学部,湖北 武汉 430079
  • 收稿日期:2023-12-06 出版日期:2025-07-15 发布日期:2025-07-14
  • 通讯作者: 陈矛
  • 基金资助:
    国家自然科学基金(62077019); 西北师范大学青年教师科研能力提升计划项目(NWNU-SKQN2023-36)

Approach for Automated Generation of Geometric Proof Problems Based on Vector Identity

LI Lei1,2, CHEN Mao2,3,*(), PENG Xicheng2,3   

  1. 1. School of Educational Technology, Northwest Normal University, Lanzhou 730070, Gansu, China
    2. National Engineering Research Center of Educational Big Data, Central China Normal University, Wuhan 430079, Hubei, China
    3. Faculty of Artificial Intelligence in Education, Central China Normal University, Wuhan 430079, Hubei, China
  • Received:2023-12-06 Online:2025-07-15 Published:2025-07-14
  • Contact: CHEN Mao

摘要:

几何证明题的自动生成是智能教育领域中的热点研究问题。现有方法通常以单道已有习题的原始几何关系为基础,利用几何自动推理技术,推理发现新的几何关系并合成新习题。由它们生成的习题所包含的几何关系全都可以基于输入习题的原始几何关系推理得到,缺乏新颖性。针对这一问题,提出一种通过重组2道已有习题的几何关系以生成新习题的方法。通过引入向量恒等式理论,为几何关系的表示以及判定源自不同习题的几何关系能否重组为新习题提供理论依据,进而实现从已有习题中自动提取几何关系,以及将源自不同习题的几何关系自动重组为新习题的算法。实验分析以及专家的评估结果表明,所提方法在运行性能以及教育应用方面都具有可行性,并且在相同输入的情况下能够生成现有方法无法生成的更具新颖性的习题。

关键词: 习题自动生成, 几何证明题, 向量恒等式, 几何定理自动发现

Abstract:

Automated generation of geometry proof problems is a prominent research area in the field of intelligent education. Current methods commonly employ automated geometric reasoning techniques to discover new geometric relationships based on the original geometric relations of an existing question, thereby synthesizing new relationships. However, the problems generated by these methods lack novelty because the geometric relationships they contain can be inferred from the original geometric relationships of the input problem. To address this limitation, this study proposes an approach for generating new problems by recombining the geometric relationships of two existing problems. By introducing the theory of vector identity equation, a theoretical foundation is established for representing geometric relationships and determining whether geometric relationships from different problems can be reorganized to form new problems. Consequently, algorithms are developed to automatically extract geometric relationships from existing problems and recombine them to generate novel problem instances. Experimental analysis and expert evaluation demonstrate the feasibility of the proposed method in terms of operational performance and educational application. Notably, under the same input conditions, the proposed method can generate more novel problems than existing methods.

Key words: automated generation of problems, geometry proof problems, vector identity equation, automated discovery of geometric theorems