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README.md

Geometry Problem Solving Agent

结合FormalGeo与Agent的几何问题形式化自动求解器。

📝 项目简介

本项目构建了一个统一的神经符号推理框架，该框架将大语言模型、智能体架构与形式化符号求解器深度融合，其中大语言模型作为规划师，负责高层次的语义理解和求解路径的反思修正，而符号求解器作为执行器，负责形式化验证与严格定理的应用执行。大语言模型的神经推理能力与符号系统的逻辑完备性互为补充，从根本上消除了模型产生幻觉的风险。此外，本项目构建了首个双向符号推理引擎，完整统一了前向推导与后向目标分解两种求解方式。该框架主要面向高精度几何定理证明与自动解题场景，可广泛应用于教育智能辅导、数学竞赛推理及几何知识验证等实际任务中。

architecture.png

✨ 核心功能

  • 神经符号协同推理：本项目构建了一套 LLM 与形式化符号求解器深度协作的 Agent 推理框架，LLM 作为“规划师”负责高层次语义理解和路径反思，符号求解器作为“执行器”负责定理的严格验证与精确计算。两者通过“推理 — 执行 — 反思 — 记忆”的闭环迭代机制协同工作，从根本上消除了大模型在长链推理中常见的幻觉问题，确保每一推导步骤均可验证、可追溯。
  • 统一双向符号推理引擎：本项目实现了首个将前向推理（从已知条件推导结论）与后向推理（从目标分解子目标）完整统一的符号求解引擎。每个定理均被定义为可逆操作，前向用于生成新事实，后向用于分解目标。系统同时从两个方向搜索，并在中间状态相遇时终止，相比单向搜索具有指数级的复杂度优势。
  • 开箱即用：本项目不依赖任何问题特定的标注数据即可直接运行于个人笔记本电脑，支持多种主流 LLM 作为后端，并提供了完整的工具调用接口（定理应用、目标分解、事实查询、状态检查等），方便开发者快速集成与二次开发。

🛠️ 技术栈

  • 使用 Hello-Agents API 实现 Reflection + ReAct + Plan-and-Solve 融合的智能体框架
  • FormalGeo 形式化系统与求解器
  • 符号计算库 SymPy

🚀 快速开始

从 Google Drive 或 百度网盘 下载数据集和日志，并将其解压到当前项目目录中。接下来在项目目录下新建一个 .env 文件。此时你的目录结构应如下所示：

BitSecret-GPSAgent/
|--datasets/
|  |--diagram/
|  |--ggbs/
|  |--problems/
|  |--summarize_prompt.txt
|  |--system_prompt.txt
|  └──gdl.json
|
|--outputs/
|  |--agent/
|  |--log/
|  |--fig-statistics.pdf
|  └──tab-main_results.txt
|
|--src/
|  └──gps/
|     |--agent_loop.py
|     |--chart.py
|     |--symbolic_solver.py
|     └──utils.py
|
|--.env
|--architecture.png
|--requirements.txt
└──README.md

新建 Python 环境并安装依赖：

$ conda create -n GPSAgent python=3.12.12
$ conda activate GPSAgent
$ pip install -r requirements.txt

运行几何问题求解器，并打印几何问题求解过程与Agent交互历史：

$ cd src/gps
$ python agent_loop.py

运行 agent_loop.py 之前，请在 RAVS 目录下添加 .env 文件，并配置以下参数：

Deepseek_BASE_URL="https://api.deepseek.com"
Deepseek_API_KEY="your_api_key"
Deepseek_MODEL_ID="deepseek-v4-pro"

如果你想使用其他基础模型，例如 Qwen3.6 Plus，可以在 .env 文件中添加以下信息，同时修改 agent_loop.py 中 main 函数的参数 model_names=['BaiLian']。如果你想使用多进程运行，可以添加多个模型名称，例如 model_names=['Deepseek', 'Deepseek', 'BaiLian']。每个模型名称对应一个进程。

BaiLian_BASE_URL="https://dashscope.aliyuncs.com/compatible-mode/v1"
BaiLian_API_KEY="your_api_key"
BaiLian_MODEL_ID="qwen3.6-plus"

📖 使用示例

运行以下代码求解几何问题：

# 示例代码
from agent_loop import main

main(
    test_pids=[1, 2, 3],  # 要运行的题目列表，范围1-7000
    log_path="../../outputs/log/log_pssr_agent.json",  # 保存日志的地址
    model_names=['Deepseek'],  # 要使用的LLM
    max_epoch=50,  # 单个问题与LLM的最大交互次数
    max_context=80000,  # 单个问题最大上下文长度
    solve_again=True,  # 设置为 True 时，若问题求解不成功时，再次运行依然尝试求解
    debug_mode=True  # 设置为 True 时输出对话交互历史
)

你将会得到类似的输出：

{
  "timing": 168.69201111793518,
  "model_name": "deepseek-v4-pro",
  "history": [
    [
      {
        "role": "system",
        "content": "<system prompt start>\n你现在作为一个研究平面几何问题的专家，使用形式化求解器FormalGeo，根据求解器的反馈，交互式的求解几何问题。形式化系统的定义、可供使用的工具、以及你的行为描述如下。\n\n\nA、形式化系统的定义\n形式化系统定义了5种概念，分别是几何结构谓词、实体、关系、属性和定理。\n\n1.几何结构谓词用来描述几何图形的拓扑结构信息，定义了图形的结构，一共有三种，分别是Shape(*)、Collinear(*)、Cocircular(*)。Shape用来描述由线或弧依次逆时针构成的图形，如Shape(AB,BC,CA)描述了一个三角形ABC、Shape(AB,BC)描述了一个角ABC、Shape(PA,OAB,BP)描述了一个扇形，P是圆心，O是圆；Collinear用来描述按顺序排列的几个共线点，如Collinear(ABC)表示点A、B和C共线；Cocircular用来描述逆时针排列的几个点共圆，如Collinear(O,XYZ)表示点X、Y和Z在圆O上，并且是逆时针顺序。\n\n2.实体是更详细的拓扑结构信息描述，求解器识别到上述3种几何结构谓词之后，会自动扩展出11种实体，分别是Point(A)、Line(A,B)、PointOnLine(M,A,B)、Angle(A,B,C)、Triangle(A,B,C)、Quadrilateral(A,B,C,D)、Circle(O)、PointOnCircle(A,O)、DoublePointsOnCircle(A,B,O)、TriplePointsOnCircle(A,B,C,O)、QuadruplePointsOnCircle(A,B,C,D,O)。实体在求解器中的作用是描述几何图形的拓扑结构信息，在添加关系时运行实体存在性检查。如添加关系RightTriangle(A,B,C)时，会检查是否存在Triangle(A,B,C)，若不存在，则实体存在性检查不通过。此外，在此形式化系统中，用点的逆时针顺序表示几何图形。以三角形为例，对于Triangle(A,B,C)，还有其他两种表示(B,C,A)和(C,A,B)；但(C,B,A)不是，(C,B,A)表示另一种镜像对称的拓扑结构。这一点在应用下文中介绍的镜像(定理名中包含mirror)对称三角形的相似和全等定理时需要注意，MirrorCongruentBetweenTriangle(A,B,C,D,E,F)表示三角形ABC和DEF镜像相似，点对应关系为A->D,B->F,C->E（而不是A->D,B->E,C->F）。\n\n3.关系用于描述实体之间的关系，如三个点的关系可能是RightTriangle(A,B,C)、一个点和一个圆的关系可能是IsCentreOfCircle(P,O)。这些关系的参数格式和它们所需的实体存在性检查为：\nRightTriangle(A,B,C):Triangle(A,B,C)\nIsoscelesTriangle(A,B,C):Triangle(A,B,C)\nIsoscelesRightTriangle(A,B,C):Triangle(A,B,C)\nEquilateralTriangle(A,B,C):Triangle(A,B,C)\nKite(A,B,C,D):Quadrilateral(A,B,C,D)\nParallelogram(A,B,C,D):Quadrilateral(A,B,C,D)\nRhombus(A,B,C,D):Quadrilateral(A,B,C,D)\nRectangle(A,B,C,D):Quadrilateral(A,B,C,D)\nSquare(A,B,C,D):Quadrilateral(A,B,C,D)\nTrapezoid(A,B,C,D):Quadrilateral(A,B,C,D)\nIsoscelesTrapezoid(A,B,C,D):Quadrilateral(A,B,C,D)\nRightTrapezoid(A,B,C,D):Quadrilateral(A,B,C,D)\nIsMidpointOfLine(M,A,B):Point(M)&Line(A,B)\nIsMidpointOfArc(M,O,A,B):Point(M)&DoublePointsOnCircle(A,B,O)\nParallelBetweenLine(A,B,C,D):Line(A,B)&Line(C,D)\nRightAngle(A,O,C):Angle(A,O,C)\nIsPerpendicularBisectorOfLine(C,O,A,B):Line(C,O)&Line(A,B)\nIsBisectorOfAngle(D,A,B,C):Line(B,D)&Angle(A,B,C)\nIsMedianOfTriangle(D,A,B,C):Line(A,D)&Triangle(A,B,C)\nIsAltitudeOfTriangle(D,A,B,C):Line(A,D)&Triangle(A,B,C)\nIsMidsegmentOfTriangle(D,E,A,B,C):Line(D,E)&Triangle(A,B,C)\nIsCircumcenterOfTriangle(O,A,B,C):Point(O)&Triangle(A,B,C)\nIsIncenterOfTriangle(O,A,B,C):Point(O)&Triangle(A,B,C)\nIsCentroidOfTriangle(O,A,B,C):Point(O)&Triangle(A,B,C)\nIsOrthocenterOfTriangle(O,A,B,C):Point(O)&Triangle(A,B,C)\nCongruentBetweenTriangle(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)\nMirrorCongruentBetweenTriangle(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)\nSimilarBetweenTriangle(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)\nMirrorSimilarBetweenTriangle(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)\nIsAltitudeOfQuadrilateral(E,F,A,B,C,D):Line(E,F)&Quadrilateral(A,B,C,D)\nIsMidsegmentOfQuadrilateral(E,F,A,B,C,D):Line(E,F)&Quadrilateral(A,B,C,D)\nIsCircumcenterOfQuadrilateral(O,A,B,C,D):Point(O)&Quadrilateral(A,B,C,D)\nIsIncenterOfQuadrilateral(O,A,B,C,D):Point(O)&Quadrilateral(A,B,C,D)\nCongruentBetweenQuadrilateral(A,B,C,D,E,F,G,H):Quadrilateral(A,B,C,D)&Quadrilateral(E,F,G,H)\nMirrorCongruentBetweenQuadrilateral(A,B,C,D,E,F,G,H):Quadrilateral(A,B,C,D)&Quadrilateral(E,F,G,H)\nSimilarBetweenQuadrilateral(A,B,C,D,E,F,G,H):Quadrilateral(A,B,C,D)&Quadrilateral(E,F,G,H)\nMirrorSimilarBetweenQuadrilateral(A,B,C,D,E,F,G,H):Quadrilateral(A,B,C,D)&Quadrilateral(E,F,G,H)\nCongruentBetweenArc(O,A,B,C,D):DoublePointsOnCircle(A,B,O)&DoublePointsOnCircle(C,D,O)\nSimilarBetweenArc(O,A,B,C,D):DoublePointsOnCircle(A,B,O)&DoublePointsOnCircle(C,D,O)\nIsDiameterOfCircle(A,B,O):Line(A,B)&DoublePointsOnCircle(A,B,O)\nIsTangentOfCircle(P,A,O):Line(P,A)&PointOnCircle(A,O)\nIsCentreOfCircle(P,O):Point(P)&Circle(O)\n\n4.除了上述定义的几何关系，形式化系统还可以表示代数关系Eq，代数关系是由属性的符号表示、运算符和数字构成的等式。如AB.ll-CD.ll=0表示线AB的长度与线CD的长度相等。属性是几何实体和关系某一性质的定量描述，如角的大小、线的长度等。属性可以使用符号表示，如AOB.ma表示角AOB的角度、AB.ll表示线AB的长度。形式化系统属性的定义和它们的符号为：\nLengthOfLine(A,B):ll\nMeasureOfAngle(A,B,C):ma\nPerimeterOfTriangle(A,B,C):pt\nAreaOfTriangle(A,B,C):at\nHeightOfTriangle(A,B,C):ht\nRatioOfSimilarTriangle(A,B,C,D,E,F):rst\nRatioOfMirrorSimilarTriangle(A,B,C,D,E,F):rmt\nPerimeterOfQuadrilateral(A,B,C,D):pq\nAreaOfQuadrilateral(A,B,C,D):aq\nHeightOfQuadrilateral(A,B,C,D):hq\nRatioOfSimilarQuadrilateral(A,B,C,D,E,F,G,H):rsq\nRatioOfMirrorSimilarQuadrilateral(A,B,C,D,E,F,G,H):rmq\nLengthOfArc(O,A,B):la\nMeasureOfArc(O,A,B):mar\nRatioOfSimilarArc(O,A,B,C,D):rsa\nRadiusOfCircle(O):rc\nDiameterOfCircle(O):dc\nPerimeterOfCircle(O):pc\nAreaOfCircle(O):ac\nPerimeterOfSector(O,A,B):ps\nAreaOfSector(O,A,B):as\n\n5.定理定义了关系之间的推理过程。一个定理由前提和结论构成，前提是由关系和逻辑连接词构成的逻辑表达式，结论是某个关系。如平行线的传递性可以定义为parallel_judgment_par_par(A,B,C,D,E,F):ParallelBetweenLine(A,B,C,D)&ParallelBetweenLine(C,D,E,F)->ParallelBetweenLine(A,B,E,F)。在应用定理时，求解器会自动进行字符替换，假设应用的定理为parallel_judgment_par_par(A,B,M,N,X,Y)，求解器会检查ParallelBetweenLine(A,B,M,N)&ParallelBetweenLine(M,N,X,Y)是否成立，如果成立，就把ParallelBetweenLine(A,B,X,Y)添加到已知条件。形式化系统定义的定理分别为：\nline_addition(A,B,C):PointOnLine(B,A,C)->Eq(Sub(AC.ll,Add(AB.ll,BC.ll)))\nmidpoint_of_line_judgment(M,A,B):PointOnLine(M,A,B)&Eq(Sub(AM.ll,MB.ll))->IsMidpointOfLine(M,A,B)\nparallel_judgment_corresponding_angle_1(A,B,C,D,E):Angle(E,A,B)&Angle(A,C,D)&PointOnLine(A,E,C)&Eq(Sub(EAB.ma,ACD.ma))->ParallelBetweenLine(A,B,C,D)\nparallel_judgment_corresponding_angle_2(A,B,C,D,E):Angle(B,A,C)&Angle(D,C,E)&PointOnLine(C,A,E)&Eq(Sub(BAC.ma,DCE.ma))->ParallelBetweenLine(A,B,C,D)\nparallel_judgment_alternate_interior_angle_1(A,B,C,D):Angle(B,A,D)&Angle(C,D,A)&Eq(Sub(BAD.ma,CDA.ma))->ParallelBetweenLine(A,B,C,D)\nparallel_judgment_alternate_interior_angle_2(A,B,C,D):Angle(C,B,A)&Angle(B,C,D)&Eq(Sub(CBA.ma,BCD.ma))->ParallelBetweenLine(A,B,C,D)\nparallel_judgment_ipsilateral_internal_angle(A,B,C,D):Angle(B,A,C)&Angle(A,C,D)&Eq(Sub(Add(BAC.ma,ACD.ma),180))->ParallelBetweenLine(A,B,C,D)\nparallel_judgment_par_par(A,B,C,D,E,F):ParallelBetweenLine(A,B,C,D)&ParallelBetweenLine(C,D,E,F)->ParallelBetweenLine(A,B,E,F)\nparallel_judgment_per_per_1(A,B,C,D):RightAngle(B,A,C)&RightAngle(A,C,D)->ParallelBetweenLine(A,B,C,D)\nparallel_property_collinear_extend_1_1(A,B,C,D,M):ParallelBetweenLine(A,B,C,D)&PointOnLine(A,M,B)->ParallelBetweenLine(M,A,C,D)\nparallel_property_collinear_extend_1_2(A,B,C,D,M):ParallelBetweenLine(A,B,C,D)&PointOnLine(A,M,B)->ParallelBetweenLine(M,B,C,D)\nparallel_property_collinear_extend_2_1(A,B,C,D,M):ParallelBetweenLine(A,B,C,D)&PointOnLine(B,A,M)->ParallelBetweenLine(A,M,C,D)\nparallel_property_collinear_extend_2_2(A,B,C,D,M):ParallelBetweenLine(A,B,C,D)&PointOnLine(B,A,M)->ParallelBetweenLine(B,M,C,D)\nparallel_property_collinear_extend_3_1(A,B,C,D,M):ParallelBetweenLine(A,B,C,D)&PointOnLine(M,A,B)->ParallelBetweenLine(A,M,C,D)\nparallel_property_collinear_extend_3_2(A,B,C,D,M):ParallelBetweenLine(A,B,C,D)&PointOnLine(A,M,B)->ParallelBetweenLine(M,B,C,D)\nparallel_property_corresponding_angle_1(A,B,C,D,E):ParallelBetweenLine(A,B,C,D)&PointOnLine(A,E,C)->Eq(Sub(EAB.ma,ACD.ma))\nparallel_property_corresponding_angle_2(A,B,C,D,E):ParallelBetweenLine(A,B,C,D)&PointOnLine(C,A,E)->Eq(Sub(BAC.ma,DCE.ma))\nparallel_property_alternate_interior_angle_1(A,B,C,D):ParallelBetweenLine(A,B,C,D)&Line(A,D)->Eq(Sub(BAD.ma,CDA.ma))\nparallel_property_alternate_interior_angle_2(A,B,C,D):ParallelBetweenLine(A,B,C,D)&Line(B,C)->Eq(Sub(CBA.ma,BCD.ma))\nparallel_property_ipsilateral_internal_angle(A,B,C,D):ParallelBetweenLine(A,B,C,D)&Line(A,C)->Eq(Sub(Add(BAC.ma,ACD.ma),180))\nperpendicular_bisector_judgment_per_and_mid(C,O,A,B):PointOnLine(O,A,B)&Angle(A,O,C)&Eq(Sub(AOC.ma,90))&Eq(Sub(AO.ll,BO.ll))->IsPerpendicularBisectorOfLine(C,O,A,B)\nperpendicular_bisector_judgment_distance_equal(C,O,A,B):PointOnLine(O,A,B)&Angle(A,O,C)&Eq(Sub(AOC.ma,90))&Eq(Sub(CA.ll,CB.ll))->IsPerpendicularBisectorOfLine(C,O,A,B)\nperpendicular_bisector_property_distance_equal(C,O,A,B):IsPerpendicularBisectorOfLine(C,O,A,B)->Eq(Sub(CA.ll,CB.ll))\nperpendicular_bisector_property_bisector(C,O,A,B):IsPerpendicularBisectorOfLine(C,O,A,B)&Angle(B,C,O)&Angle(O,C,A)->IsBisectorOfAngle(O,B,C,A)\nangle_addition(A,B,C,D):Angle(A,B,C)&Angle(C,B,D)&Angle(A,B,D)->Eq(Sub(ABD.ma,Add(ABC.ma,CBD.ma)))\nflat_angle(A,B,C):PointOnLine(B,A,C)->Eq(Sub(ABC.ma,180))\nadjacent_complementary_angle(A,O,B,C):PointOnLine(O,A,C)&Angle(A,O,B)&Angle(B,O,C)->Eq(Sub(Add(AOB.ma,BOC.ma),180))\nround_angle(A,O,B):Angle(A,O,B)&Angle(B,O,A)->Eq(Sub(Add(AOB.ma,BOA.ma),360))\nvertical_angle(A,O,C,B,D):PointOnLine(O,A,B)&PointOnLine(O,C,D)&Angle(A,O,C)&Angle(B,O,D)->Eq(Sub(AOC.ma,BOD.ma))\nbisector_of_angle_judgment_angle_equal(D,A,B,C):Angle(A,B,D)&Angle(D,B,C)&Eq(Sub(ABD.ma,DBC.ma))->IsBisectorOfAngle(D,A,B,C)\nbisector_of_angle_property_distance_equal(D,A,B,C):IsBisectorOfAngle(D,A,B,C)&Eq(Sub(BCD.ma,90))&Eq(Sub(DAB.ma,90))->Eq(Sub(DA.ll,DC.ll))\nbisector_of_angle_property_line_ratio(D,A,B,C):IsBisectorOfAngle(D,A,B,C)&PointOnLine(D,C,A)->Eq(Sub(Add(Log(CD.ll),Log(BA.ll)),Add(Log(DA.ll),Log(BC.ll))))\ntriangle_property_angle_sum(A,B,C):Triangle(A,B,C)->Eq(Sub(Add(ABC.ma,BCA.ma,CAB.ma),180))\ntriangle_perimeter_formula(A,B,C):Triangle(A,B,C)->Eq(Sub(ABC.pt,Add(AB.ll,BC.ll,CA.ll)))\ntriangle_area_formula_common(A,B,C):Triangle(A,B,C)->Eq(Sub(ABC.at,Mul(ABC.ht,BC.ll,1/2)))\ntriangle_area_formula_right_triangle(A,B,C):RightTriangle(A,B,C)->Eq(Sub(ABC.at,Mul(AB.ll,BC.ll,1/2)))\nmedian_of_triangle_judgment(D,A,B,C):Triangle(A,B,C)&Line(A,D)&PointOnLine(D,B,C)&Eq(Sub(BD.ll,CD.ll))->IsMedianOfTriangle(D,A,B,C)\naltitude_of_triangle_judgment_1(D,A,B,C):Triangle(A,B,C)&Line(A,D)&PointOnLine(D,B,C)&Eq(Sub(BDA.ma,90))->IsAltitudeOfTriangle(D,A,B,C)\naltitude_of_triangle_judgment_2(D,A,B,C):Triangle(A,B,C)&Line(A,D)&PointOnLine(B,D,C)&Eq(Sub(ADB.ma,90))->IsAltitudeOfTriangle(D,A,B,C)\nmidsegment_of_triangle_judgment_midpoint(D,E,A,B,C):Triangle(A,B,C)&PointOnLine(D,A,B)&PointOnLine(E,A,C)&Line(D,E)&Eq(Sub(AD.ll,BD.ll))&Eq(Sub(AE.ll,CE.ll))->IsMidsegmentOfTriangle(D,E,A,B,C)\nmidsegment_of_triangle_judgment_parallel_1(D,E,A,B,C):Triangle(A,B,C)&PointOnLine(D,A,B)&PointOnLine(E,A,C)&Line(D,E)&ParallelBetweenLine(D,E,B,C)&Eq(Sub(AD.ll,BD.ll))->IsMidsegmentOfTriangle(D,E,A,B,C)\nmidsegment_of_triangle_judgment_parallel_2(D,E,A,B,C):Triangle(A,B,C)&PointOnLine(D,A,B)&PointOnLine(E,A,C)&Line(D,E)&ParallelBetweenLine(D,E,B,C)&Eq(Sub(AE.ll,CE.ll))->IsMidsegmentOfTriangle(D,E,A,B,C)\nmidsegment_of_triangle_property_parallel(D,E,A,B,C):IsMidsegmentOfTriangle(D,E,A,B,C)->ParallelBetweenLine(D,E,B,C)\nmidsegment_of_triangle_property_length(D,E,A,B,C):IsMidsegmentOfTriangle(D,E,A,B,C)->Eq(Sub(DE.ll,Mul(BC.ll,1/2)))\ncentroid_of_triangle_judgment_intersection(O,A,B,C,M,N):IsMedianOfTriangle(M,C,A,B)&IsMedianOfTriangle(N,B,C,A)&PointOnLine(O,C,M)&PointOnLine(O,B,N)->IsCentroidOfTriangle(O,A,B,C)\ncentroid_of_triangle_property_intersection(O,A,B,C,M):IsCentroidOfTriangle(O,A,B,C)&PointOnLine(O,A,M)&PointOnLine(M,B,C)->IsMedianOfTriangle(M,A,B,C)\ncentroid_of_triangle_property_line_ratio(O,A,B,C,M):IsCentroidOfTriangle(O,A,B,C)&PointOnLine(O,A,M)&PointOnLine(M,B,C)->Eq(Sub(OA.ll,Mul(OM.ll,2)))\ncongruent_triangle_judgment_aas_1(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,DEF.ma))&Eq(Sub(BCA.ma,EFD.ma))&Eq(Sub(AB.ll,DE.ll))->CongruentBetweenTriangle(A,B,C,D,E,F)\ncongruent_triangle_judgment_aas_2(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,DEF.ma))&Eq(Sub(BCA.ma,EFD.ma))&Eq(Sub(BC.ll,EF.ll))->CongruentBetweenTriangle(A,B,C,D,E,F)\ncongruent_triangle_judgment_hl_2(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,90))&Eq(Sub(DEF.ma,90))&Eq(Sub(AC.ll,DF.ll))&Eq(Sub(BC.ll,EF.ll))->CongruentBetweenTriangle(A,B,C,D,E,F)\ncongruent_triangle_property_line_equal(A,B,C,D,E,F):CongruentBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(BC.ll,EF.ll))\ncongruent_triangle_property_angle_equal(A,B,C,D,E,F):CongruentBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(CAB.ma,FDE.ma))\nmirror_congruent_triangle_judgment_sss(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(AB.ll,FD.ll))&Eq(Sub(BC.ll,EF.ll))&Eq(Sub(CA.ll,DE.ll))->MirrorCongruentBetweenTriangle(A,B,C,D,E,F)\nmirror_congruent_triangle_judgment_sas(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(AB.ll,DF.ll))&Eq(Sub(CAB.ma,FDE.ma))&Eq(Sub(AC.ll,DE.ll))->MirrorCongruentBetweenTriangle(A,B,C,D,E,F)\nmirror_congruent_triangle_judgment_aas_1(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,EFD.ma))&Eq(Sub(BCA.ma,DEF.ma))&Eq(Sub(AB.ll,DF.ll))->MirrorCongruentBetweenTriangle(A,B,C,D,E,F)\nmirror_congruent_triangle_judgment_aas_2(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,EFD.ma))&Eq(Sub(BCA.ma,DEF.ma))&Eq(Sub(BC.ll,EF.ll))->MirrorCongruentBetweenTriangle(A,B,C,D,E,F)\nmirror_congruent_triangle_judgment_aas_3(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,EFD.ma))&Eq(Sub(BCA.ma,DEF.ma))&Eq(Sub(CA.ll,DE.ll))->MirrorCongruentBetweenTriangle(A,B,C,D,E,F)\nmirror_congruent_triangle_judgment_hl_1(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,90))&Eq(Sub(EFD.ma,90))&Eq(Sub(AC.ll,DE.ll))&Eq(Sub(BC.ll,EF.ll))->MirrorCongruentBetweenTriangle(A,B,C,D,E,F)\nmirror_congruent_triangle_judgment_hl_2(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,90))&Eq(Sub(EFD.ma,90))&Eq(Sub(AC.ll,DE.ll))&Eq(Sub(AB.ll,DF.ll))->MirrorCongruentBetweenTriangle(A,B,C,D,E,F)\nmirror_congruent_triangle_property_line_equal(A,B,C,D,E,F):MirrorCongruentBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(BC.ll,EF.ll))\nmirror_congruent_triangle_property_angle_equal(A,B,C,D,E,F):MirrorCongruentBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(CAB.ma,FDE.ma))\nsimilar_triangle_judgment_sas(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(Add(Log(AB.ll),Log(DF.ll)),Add(Log(DE.ll),Log(AC.ll))))&Eq(Sub(CAB.ma,FDE.ma))->SimilarBetweenTriangle(A,B,C,D,E,F)\nsimilar_triangle_judgment_aa(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,DEF.ma))&Eq(Sub(BCA.ma,EFD.ma))->SimilarBetweenTriangle(A,B,C,D,E,F)\nsimilar_triangle_property_line_ratio(A,B,C,D,E,F):SimilarBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(Log(BC.ll),Add(Log(EF.ll),Log(ABCDEF.rst))))\nsimilar_triangle_property_angle_equal(A,B,C,D,E,F):SimilarBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(CAB.ma,FDE.ma))\nsimilar_triangle_property_perimeter_ratio(A,B,C,D,E,F):SimilarBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(Log(ABC.pt),Add(Log(DEF.pt),Log(ABCDEF.rst))))\nsimilar_triangle_property_area_square_ratio(A,B,C,D,E,F):SimilarBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(Log(ABC.at),Add(Log(DEF.at),Log(ABCDEF.rst),Log(ABCDEF.rst))))\nmirror_similar_triangle_judgment_sas(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(Add(Log(AB.ll),Log(DE.ll)),Add(Log(DF.ll),Log(AC.ll))))&Eq(Sub(CAB.ma,FDE.ma))->MirrorSimilarBetweenTriangle(A,B,C,D,E,F)\nmirror_similar_triangle_judgment_aa(A,B,C,D,E,F):Triangle(A,B,C)&Triangle(D,E,F)&Eq(Sub(ABC.ma,EFD.ma))&Eq(Sub(BCA.ma,DEF.ma))->MirrorSimilarBetweenTriangle(A,B,C,D,E,F)\nmirror_similar_triangle_property_line_ratio(A,B,C,D,E,F):MirrorSimilarBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(Log(BC.ll),Add(Log(EF.ll),Log(ABCDEF.rmt))))\nmirror_similar_triangle_property_angle_equal(A,B,C,D,E,F):MirrorSimilarBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(CAB.ma,FDE.ma))\nmirror_similar_triangle_property_area_square_ratio(A,B,C,D,E,F):MirrorSimilarBetweenTriangle(A,B,C,D,E,F)->Eq(Sub(Log(ABC.at),Add(Log(DEF.at),Log(ABCDEF.rmt),Log(ABCDEF.rmt))))\nright_triangle_judgment_angle(A,B,C):Triangle(A,B,C)&Eq(Sub(ABC.ma,90))->RightTriangle(A,B,C)\nright_triangle_judgment_pythagorean_inverse(A,B,C):Triangle(A,B,C)&Eq(Sub(Add(Pow(AB.ll,2),Pow(BC.ll,2)),Pow(AC.ll,2)))->RightTriangle(A,B,C)\nright_triangle_property_pythagorean(A,B,C):RightTriangle(A,B,C)->Eq(Sub(Add(Pow(AB.ll,2),Pow(BC.ll,2)),Pow(AC.ll,2)))\nright_triangle_property_length_of_median(A,B,C,M):RightTriangle(A,B,C)&IsMedianOfTriangle(M,B,C,A)->Eq(Sub(Mul(BM.ll,2),CA.ll))\nright_triangle_property_special_angle_1(A,B,C):RightTriangle(A,B,C)&Eq(Sub(CAB.ma,30))->Eq(Sub(AC.ll,Mul(BC.ll,2)))\nright_triangle_property_special_angle_2(A,B,C):RightTriangle(A,B,C)&Eq(Sub(CAB.ma,60))->Eq(Sub(AC.ll,Mul(AB.ll,2)))\nisosceles_triangle_judgment_line_equal(A,B,C):Triangle(A,B,C)&Eq(Sub(AB.ll,AC.ll))->IsoscelesTriangle(A,B,C)\nisosceles_triangle_judgment_angle_equal(A,B,C):Triangle(A,B,C)&Eq(Sub(ABC.ma,BCA.ma))->IsoscelesTriangle(A,B,C)\nisosceles_triangle_property_angle_equal(A,B,C):IsoscelesTriangle(A,B,C)->Eq(Sub(ABC.ma,BCA.ma))\nisosceles_triangle_property_line_coincidence_1_1(A,B,C,M):IsoscelesTriangle(A,B,C)&IsAltitudeOfTriangle(M,A,B,C)->IsMedianOfTriangle(M,A,B,C)\nisosceles_triangle_property_line_coincidence_2_1(A,B,C,M):IsoscelesTriangle(A,B,C)&IsMedianOfTriangle(M,A,B,C)->IsAltitudeOfTriangle(M,A,B,C)\nisosceles_triangle_property_line_coincidence_2_2(A,B,C,M):IsoscelesTriangle(A,B,C)&IsMedianOfTriangle(M,A,B,C)->IsBisectorOfAngle(M,C,A,B)\nequilateral_triangle_judgment_isosceles_and_isosceles(A,B,C):IsoscelesTriangle(A,B,C)&IsoscelesTriangle(B,C,A)->EquilateralTriangle(A,B,C)\nequilateral_triangle_property_angle(A,B,C):EquilateralTriangle(A,B,C)->Eq(Sub(CAB.ma,60))\nquadrilateral_property_angle_sum(A,B,C,D):Quadrilateral(A,B,C,D)->Eq(Sub(Add(ABC.ma,BCD.ma,CDA.ma,DAB.ma),360))\nquadrilateral_perimeter_formula(A,B,C,D):Quadrilateral(A,B,C,D)->Eq(Sub(Add(AB.ll,BC.ll,CD.ll,DA.ll),ABCD.pq))\naltitude_of_quadrilateral_judgment_left_vertex_1(F,A,B,C,D):Parallelogram(A,B,C,D)&Line(A,F)&PointOnLine(F,B,C)&Eq(Sub(BFA.ma,90))->IsAltitudeOfQuadrilateral(A,F,A,B,C,D)\naltitude_of_quadrilateral_judgment_left_vertex_3(F,A,B,C,D):Parallelogram(A,B,C,D)&Line(A,F)&PointOnLine(B,F,C)&Eq(Sub(AFB.ma,90))->IsAltitudeOfQuadrilateral(A,F,A,B,C,D)\naltitude_of_quadrilateral_judgment_right_vertex_1(F,A,B,C,D):Parallelogram(A,B,C,D)&Line(D,F)&PointOnLine(F,B,C)&Eq(Sub(BFD.ma,90))->IsAltitudeOfQuadrilateral(D,F,A,B,C,D)\naltitude_of_quadrilateral_judgment_right_vertex_2(F,A,B,C,D):Trapezoid(A,B,C,D)&Line(D,F)&PointOnLine(F,B,C)&Eq(Sub(BFD.ma,90))->IsAltitudeOfQuadrilateral(D,F,A,B,C,D)\naltitude_of_quadrilateral_judgment_right_vertex_5(F,A,B,C,D):Parallelogram(A,B,C,D)&Line(D,F)&PointOnLine(C,B,F)&Eq(Sub(CFD.ma,90))->IsAltitudeOfQuadrilateral(D,F,A,B,C,D)\naltitude_of_quadrilateral_judgment_diagonal_1(A,B,C,D):Parallelogram(A,B,C,D)&Line(A,C)&Eq(Sub(BCA.ma,90))->IsAltitudeOfQuadrilateral(A,C,A,B,C,D)\naltitude_of_quadrilateral_judgment_diagonal_3(A,B,C,D):Parallelogram(A,B,C,D)&Line(D,B)&Eq(Sub(DBC.ma,90))->IsAltitudeOfQuadrilateral(D,B,A,B,C,D)\nmidsegment_of_quadrilateral_judgment_midpoint(E,F,A,B,C,D):PointOnLine(E,A,B)&PointOnLine(F,D,C)&Line(E,F)&Eq(Sub(AE.ll,BE.ll))&Eq(Sub(DF.ll,CF.ll))->IsMidsegmentOfQuadrilateral(E,F,A,B,C,D)\nmidsegment_of_quadrilateral_judgment_parallel_1(E,F,A,B,C,D):PointOnLine(E,A,B)&PointOnLine(F,D,C)&Line(E,F)&Trapezoid(A,B,C,D)&ParallelBetweenLine(E,F,B,C)&Eq(Sub(AE.ll,BE.ll))->IsMidsegmentOfQuadrilateral(E,F,A,B,C,D)\nmidsegment_of_quadrilateral_property_length(E,F,A,B,C,D):IsMidsegmentOfQuadrilateral(E,F,A,B,C,D)->Eq(Sub(Add(AD.ll,BC.ll),Mul(EF.ll,2)))\nmirror_congruent_quadrilateral_property_angle_equal(A,B,C,D,E,F,G,H):MirrorCongruentBetweenQuadrilateral(A,B,C,D,E,F,G,H)->Eq(Sub(DAB.ma,HEF.ma))\nsimilar_quadrilateral_property_line_ratio(A,B,C,D,E,F,G,H):SimilarBetweenQuadrilateral(A,B,C,D,E,F,G,H)->Eq(Sub(Log(AB.ll),Add(Log(EF.ll),Log(ABCDEFGH.rsq))))\nsimilar_quadrilateral_property_area_square_ratio(A,B,C,D,E,F,G,H):SimilarBetweenQuadrilateral(A,B,C,D,E,F,G,H)->Eq(Sub(Log(ABCD.aq),Add(Log(EFGH.aq),Log(ABCDEFGH.rsq),Log(ABCDEFGH.rsq))))\nparallelogram_judgment_parallel_and_parallel(A,B,C,D):Quadrilateral(A,B,C,D)&ParallelBetweenLine(A,D,B,C)&ParallelBetweenLine(B,A,C,D)->Parallelogram(A,B,C,D)\nparallelogram_judgment_equal_and_equal(A,B,C,D):Quadrilateral(A,B,C,D)&Eq(Sub(AD.ll,BC.ll))&Eq(Sub(BA.ll,CD.ll))->Parallelogram(A,B,C,D)\nparallelogram_judgment_angle_and_angle(A,B,C,D):Quadrilateral(A,B,C,D)&Eq(Sub(DAB.ma,BCD.ma))&Eq(Sub(ABC.ma,CDA.ma))->Parallelogram(A,B,C,D)\nparallelogram_property_opposite_line_equal(A,B,C,D):Parallelogram(A,B,C,D)->Eq(Sub(BA.ll,CD.ll))\nparallelogram_property_opposite_angle_equal(A,B,C,D):Parallelogram(A,B,C,D)->Eq(Sub(DAB.ma,BCD.ma))\nparallelogram_property_diagonal_bisection(A,B,C,D,O):Parallelogram(A,B,C,D)&PointOnLine(O,A,C)&PointOnLine(O,B,D)->IsMidpointOfLine(O,A,C)\nparallelogram_area_formula_common(A,B,C,D):Parallelogram(A,B,C,D)->Eq(Sub(Log(ABCD.aq),Add(Log(ABCD.hq),Log(BC.ll))))\nrectangle_area_formula(A,B,C,D):Rectangle(A,B,C,D)->Eq(Sub(Log(ABCD.aq),Add(Log(AB.ll),Log(BC.ll))))\nkite_judgment_equal_and_equal(A,B,C,D):Quadrilateral(A,B,C,D)&Eq(Sub(AB.ll,AD.ll))&Eq(Sub(CB.ll,CD.ll))->Kite(A,B,C,D)\nkite_property_diagonal_perpendicular_bisection(A,B,C,D,O):Kite(A,B,C,D)&PointOnLine(O,A,C)&PointOnLine(O,B,D)->IsPerpendicularBisectorOfLine(A,O,B,D)\nkite_property_opposite_angle_equal(A,B,C,D):Kite(A,B,C,D)->Eq(Sub(ABC.ma,CDA.ma))\nkite_area_formula_diagonal(A,B,C,D):Kite(A,B,C,D)&Line(B,D)&Line(A,C)->Eq(Sub(ABCD.aq,Mul(BD.ll,AC.ll,1/2)))\nrectangle_judgment_right_angle(A,B,C,D):Parallelogram(A,B,C,D)&Eq(Sub(ABC.ma,90))->Rectangle(A,B,C,D)\nrectangle_property_diagonal_equal(A,B,C,D):Rectangle(A,B,C,D)&Line(A,C)&Line(B,D)->Eq(Sub(AC.ll,BD.ll))\nrhombus_judgment_parallelogram_and_kite(A,B,C,D):Parallelogram(A,B,C,D)&Kite(A,B,C,D)->Rhombus(A,B,C,D)\ntrapezoid_judgment_parallel(A,B,C,D):Quadrilateral(A,B,C,D)&ParallelBetweenLine(A,D,B,C)->Trapezoid(A,B,C,D)\ntrapezoid_area_formula(A,B,C,D):Trapezoid(A,B,C,D)->Eq(Sub(ABCD.aq,Mul(Add(AD.ll,BC.ll),ABCD.hq,1/2)))\nright_trapezoid_judgment_right_angle(A,B,C,D):Trapezoid(A,B,C,D)&Eq(Sub(ABC.ma,90))->RightTrapezoid(A,B,C,D)\nright_trapezoid_area_formular(A,B,C,D):RightTrapezoid(A,B,C,D)->Eq(Sub(ABCD.aq,Mul(Add(AD.ll,BC.ll),AB.ll,1/2)))\nisosceles_trapezoid_judgment_line_equal(A,B,C,D):Trapezoid(A,B,C,D)&Eq(Sub(AB.ll,CD.ll))->IsoscelesTrapezoid(A,B,C,D)\nisosceles_trapezoid_property_diagonal_equal(A,B,C,D):IsoscelesTrapezoid(A,B,C,D)->Eq(Sub(AC.ll,BD.ll))\nround_arc(O,A,B):DoublePointsOnCircle(A,B,O)->Eq(Sub(Add(OAB.mar,OBA.mar),360))\narc_addition_length(O,A,B,C):TriplePointsOnCircle(A,B,C,O)->Eq(Sub(OAC.la,Add(OAB.la,OBC.la)))\narc_addition_measure(O,A,B,C):TriplePointsOnCircle(A,B,C,O)->Eq(Sub(OAC.mar,Add(OAB.mar,OBC.mar)))\narc_property_center_angle(O,A,B,P):DoublePointsOnCircle(A,B,O)&Angle(B,P,A)&IsCentreOfCircle(P,O)->Eq(Sub(OAB.mar,BPA.ma))\narc_property_circumference_angle_external(O,A,B,C):TriplePointsOnCircle(A,B,C,O)&Angle(B,C,A)->Eq(Sub(BCA.ma,Mul(OAB.mar,1/2)))\narc_property_circumference_angle_internal(O,A,B,D):TriplePointsOnCircle(A,D,B,O)&Angle(A,D,B)->Eq(Sub(ADB.ma,Sub(180,Mul(OAB.mar,1/2))))\narc_length_formula(O,A,B):DoublePointsOnCircle(A,B,O)->Eq(Sub(OAB.la,Mul(OAB.mar,1/180*pi,O.rc)))\ncongruent_arc_judgment_length_equal(O,A,B,C,D):DoublePointsOnCircle(A,B,O)&DoublePointsOnCircle(C,D,O)&Eq(Sub(OAB.la,OCD.la))->CongruentBetweenArc(O,A,B,C,D)\ncongruent_arc_judgment_measure_equal(O,A,B,C,D):DoublePointsOnCircle(A,B,O)&DoublePointsOnCircle(C,D,O)&Eq(Sub(OAB.mar,OCD.mar))->CongruentBetweenArc(O,A,B,C,D)\ncongruent_arc_judgment_chord_equal(O,A,B,C,D):DoublePointsOnCircle(A,B,O)&DoublePointsOnCircle(C,D,O)&Line(A,B)&Line(C,D)&Eq(Sub(AB.ll,CD.ll))->CongruentBetweenArc(O,A,B,C,D)\ncongruent_arc_property_length_equal(O,A,B,C,D):CongruentBetweenArc(O,A,B,C,D)->Eq(Sub(OAB.la,OCD.la))\ncongruent_arc_property_measure_equal(O,A,B,C,D):CongruentBetweenArc(O,A,B,C,D)->Eq(Sub(OAB.mar,OCD.mar))\ncongruent_arc_property_chord_equal(O,A,B,C,D):CongruentBetweenArc(O,A,B,C,D)&Line(A,B)&Line(C,D)->Eq(Sub(AB.ll,CD.ll))\nsimilar_arc_judgment_cocircular(O,A,B,C,D):DoublePointsOnCircle(A,B,O)&DoublePointsOnCircle(C,D,O)->SimilarBetweenArc(O,A,B,C,D)\nsimilar_arc_property_length_ratio(O,A,B,C,D):SimilarBetweenArc(O,A,B,C,D)->Eq(Sub(Log(OAB.la),Add(Log(OCD.la),Log(OCDAB.rsa))))\nsimilar_arc_property_measure_ratio(O,A,B,C,D):SimilarBetweenArc(O,A,B,C,D)->Eq(Sub(Log(OAB.mar),Add(Log(OCD.mar),Log(OCDAB.rsa))))\ncircle_property_length_of_radius_and_diameter(O):Circle(O)->Eq(Sub(O.dc,Mul(O.rc,2)))\ncircle_property_circular_power_chord_and_chord(A,E,B,C,D,O):DoublePointsOnCircle(A,B,O)&DoublePointsOnCircle(C,D,O)&PointOnLine(E,A,B)&PointOnLine(E,C,D)->Eq(Sub(Add(Log(EC.ll),Log(ED.ll)),Add(Log(EA.ll),Log(EB.ll))))\ncircle_property_circular_power_tangent_and_segment_line(P,A,C,D,O):IsTangentOfCircle(P,A,O)&DoublePointsOnCircle(C,D,O)&PointOnLine(C,P,D)->Eq(Sub(Add(Log(PA.ll),Log(PA.ll)),Add(Log(PC.ll),Log(PD.ll))))\ncircle_property_circular_power_segment_and_segment_line(P,A,B,C,D,O):DoublePointsOnCircle(A,B,O)&DoublePointsOnCircle(C,D,O)&PointOnLine(A,P,B)&PointOnLine(C,P,D)->Eq(Sub(Add(Log(PA.ll),Log(PB.ll)),Add(Log(PC.ll),Log(PD.ll))))\ncircle_property_circular_power_tangent_and_segment_angle_1(P,A,C,D,O):TriplePointsOnCircle(A,C,D,O)&PointOnLine(C,P,D)->Eq(Sub(Sub(ODA.mar,OAC.mar),Mul(APC.ma,2)))\ncircle_property_circular_power_tangent_and_segment_angle_2(P,A,C,D,O):TriplePointsOnCircle(C,A,D,O)&PointOnLine(C,P,D)->Eq(Sub(Sub(OAD.mar,OCA.mar),Mul(CPA.ma,2)))\ncircle_property_circular_power_segment_and_segment_angle_1(P,A,B,C,D,O):QuadruplePointsOnCircle(A,C,D,B,O)&PointOnLine(A,P,B)&PointOnLine(C,P,D)->Eq(Sub(Sub(ODB.mar,OAC.mar),Mul(APC.ma,2)))\ncircle_property_chord_perpendicular_bisect_chord_1(O,P,M,A,B):DoublePointsOnCircle(A,B,O)&PointOnLine(M,A,B)&IsCentreOfCircle(P,O)&Eq(Sub(AMP.ma,90))->IsPerpendicularBisectorOfLine(P,M,A,B)\ncircle_property_chord_perpendicular_bisect_chord_2(O,P,M,A,B):DoublePointsOnCircle(A,B,O)&PointOnLine(M,A,B)&IsCentreOfCircle(P,O)&Angle(A,M,P)&Eq(Sub(AM.ll,MB.ll))->IsPerpendicularBisectorOfLine(P,M,A,B)\ncircle_property_chord_perpendicular_bisect_arc_1(O,A,B,P,M,D):TriplePointsOnCircle(A,D,B,O)&PointOnLine(M,A,B)&PointOnLine(M,P,D)&IsCentreOfCircle(P,O)&Eq(Sub(AMP.ma,90))->Eq(Sub(OAD.la,ODB.la))\ncircle_property_angle_of_osculation_2(O,A,B,P):DoublePointsOnCircle(A,B,O)&Angle(P,B,A)&IsTangentOfCircle(P,B,O)->Eq(Sub(PBA.ma,Mul(OAB.mar,1/2)))\ncircle_perimeter_formula(O):Circle(O)->Eq(Sub(O.pc,Mul(2*pi,O.rc)))\ncircle_area_formula(O):Circle(O)->Eq(Sub(O.ac,Mul(pi,O.rc,O.rc)))\nradius_of_circle_property_length_equal(P,A,O):PointOnCircle(A,O)&Line(P,A)&IsCentreOfCircle(P,O)->Eq(Sub(PA.ll,O.rc))\ndiameter_of_circle_judgment_pass_centre(A,P,B,O):DoublePointsOnCircle(A,B,O)&PointOnLine(P,A,B)&IsCentreOfCircle(P,O)->IsDiameterOfCircle(A,B,O)\ndiameter_of_circle_judgment_right_angle(B,C,A,O):TriplePointsOnCircle(B,C,A,O)&Eq(Sub(BCA.ma,90))->IsDiameterOfCircle(A,B,O)\ndiameter_of_circle_property_length_equal(A,B,O):IsDiameterOfCircle(A,B,O)->Eq(Sub(AB.ll,O.dc))\ndiameter_of_circle_property_right_angle(B,C,A,O):IsDiameterOfCircle(A,B,O)&TriplePointsOnCircle(B,C,A,O)&Angle(B,C,A)->RightAngle(B,C,A)\ntangent_of_circle_judgment_perpendicular_1(P,A,O,Q):PointOnCircle(A,O)&IsCentreOfCircle(Q,O)&Angle(Q,A,P)&Eq(Sub(QAP.ma,90))->IsTangentOfCircle(P,A,O)\ntangent_of_circle_property_perpendicular_1(P,A,O,Q):IsTangentOfCircle(P,A,O)&Angle(Q,A,P)&IsCentreOfCircle(Q,O)->RightAngle(Q,A,P)\ntangent_of_circle_property_perpendicular_2(P,A,O,Q):IsTangentOfCircle(P,A,O)&Angle(P,A,Q)&IsCentreOfCircle(Q,O)->RightAngle(P,A,Q)\ntangent_of_circle_property_length_equal(P,A,B,O):IsTangentOfCircle(P,A,O)&IsTangentOfCircle(P,B,O)->Eq(Sub(PA.ll,PB.ll))\nsector_area_formula(O,A,B):DoublePointsOnCircle(A,B,O)->Eq(Sub(OAB.as,Mul(OAB.mar,1/360*pi,O.rc,O.rc)))\n\n\nB、可使用的工具\n你可以使用工具辅助你求解几何问题，现在系统内定义了6种工具，实现与求解器的交互。工具的使用方法和作用描述如下：\n\n1.apply(theorem)。尝试应用一条定理theorem扩展已知条件，并返回应用结果。如应用平行线的传递性定理，可以表示为apply(parallel_judgment_par_par())，推理器会自动组合所有相关前提得到对应结论。成功应用定理后，返回应用当前定理添加的新条件；应用定理不成功，则返回失败原因（如某个前提不存在）；也可能返回报错信息，如定理未定义等。此外对于某些定理，必须使用带参数的定理形式，这些定理为bisector_of_angle_property_line_ratio、right_triangle_property_pythagorean、circle_property_circular_power_chord_and_chord、circle_property_circular_power_tangent_and_segment_line、circle_property_circular_power_segment_and_segment_line、以及求解周长（perimeter）和面积（area）相关的定理、相似（similar）的判定和性质定理、全等（congruent）的判定和性质定理。例如想要求解三角形ABC的面积，需调用工具apply(triangle_area_formula_common(A,B,C))。除这些定理外，对于其他定理，强烈建议使用定理的无参模式。\n\n2.decompose(theorem)。尝试使用一条定理theorem分解某个目标，并返回分解结果。这个工具可以将定理的结论分解为他的前提，如分解平行线的传递性定理，decompose(parallel_judgment_par_par(A,B,M,N,X,Y))，若存在目标ParallelBetweenLine(A,B,X,Y)，推理器会将其分解为ParallelBetweenLine(A,B,M,N)和ParallelBetweenLine(M,N,X,Y)两个子目标。与apply(theorem)工具不同，decompose(theorem)只能使用定理的带参数形式。成功使用定理分解目标后，返回新分解的子目标；使用定理分解目标不成功，则返回失败原因（如待分解的目标不存在）；也可能返回报错信息，如定理未定义等。\n\n3.find_fact(relation_type)。返回所有种类为relation的条件。例如调用find(ParallelBetweenLine)，可能会返回(A,B,C,D), (C,D,E,F)等；也可能返回报错信息，如关系未定义等。\n\n4.find_goal(relation_type)。返回所有种类为relation的目标。例如调用find(ParallelBetweenLine)，可能会返回(A,B,X,Y)；也可能返回报错信息，如关系未定义等。\n\n5.check()。返回当前几何问题的目标是否被成功求解。你可以使用这个工具，判断自己的求解过程是否要终止。特别要注意只有check()的返回结果才可以作为求解过程是否要终止的依据，解题过程需要经过形式化系统的验证。\n\n6.finish()。当你调用check()发现当前几何问题的求解目标已成功求解，简单总结求解过程，并调用finish()结束求解。\n\n\nC、你的行为描述\n你现在作为一个研究平面几何问题的专家，使用形式化求解器FormalGeo，根据求解器的反馈，交互式的求解几何问题。只有经过求解器一步步验证，调用check()工具，得到的反馈信息是问题已经求解时，才算求解完成。求解几何问题时，可以采取两种思路。第一种思路是前向求解，根据问题的已知条件，使用定理推导出新的条件，直至目标被推导出；第二种是后向求解，从问题的目标出发，使用定理将其分解为子目标，直至某个解题路径的子目标全部已知。这两种思路可以在解题过程中混合使用，即调用上述工具apply(theorem)和decompose(theorem)。当你在某一步制定计划后，后续的步骤尽量按照计划一步步执行，如果中途发现部分计划无法实现，你需要重新调整计划。你的输出必须是以下JSON格式：\n\n{\n    \"thinking\":\"你的思考过程和下一步计划\",\n    \"action\":\"你希望调用的工具\"\n}\n\nthinking的内容是你的思考过程和下一步计划。action的内容是你希望调用的工具，严格按照要求的格式调用。\n\n\nD、输出示例\n以下是几个输出示例，你必须严格按照此JSON格式输出（thinking的内容只是提供示例，不要求与其相同的句式）。\n\n示例1：\n{\n    \"thinking\":\"这个几何问题要求得到三角形的面积，已知...定理的前提都已满足，可以调用apply工具推出结论。面积相关的定理需要加定理参数。\",\n    \"action\":\"apply(triangle_area_formula_common(A,B,C))\"\n}\n\n示例2：\n{\n    \"thinking\":\"这个几何问题要求推导出平行关系，...调用apply工具，使用定理的无参数形式，推出所有平行相关的结论。\",\n    \"action\":\"apply(parallel_judgment_par_par())\"\n}\n\n示例3：\n{\n    \"thinking\":\"这个几何问题要求推导...暂且无求解思路，可以尝试将目标分解。\",\n    \"action\":\"decompose(midpoint_of_line_judgment(M,A,B))\"\n}\n\n示例4：\n{\n    \"thinking\":\"上述解题过程已经推导出...在应用定理前，我需要查看当前已知条件存在哪些代数关系。\",\n    \"action\":\"find_fact(Eq)\"\n}\n\n示例5：\n{\n    \"thinking\":\"已知目标...在应用定理前，我需要查看当前需要求解的平行关系目标有哪些。\",\n    \"action\":\"find_goal(ParallelBetweenLine)\"\n}\n\n示例6：\n{\n    \"thinking\":\"通过上述解题过程，可以...我需要验证问题是否已经成功求解。\",\n    \"action\":\"check()\"\n}\n\n示例7：\n{\n    \"thinking\":\"调用check()工具后，已确认问题已经成功求解，结束解题。\",\n    \"action\":\"finish()\"\n}\n<system prompt end>"
      },
      {
        "role": "user",
        "content": "当前问题的状态描述如下所示：\n几何图形的结构信息描述：\nShape(RS,ST,TR), Shape(XY,YZ,ZX)\n几何问题的初始已知条件：\nCongruentBetweenTriangle(R,S,T,X,Y,Z), Eq(TR.ll-x-21), Eq(ZX.ll-2*x+14), Eq(TRS.ma-4*y+10), Eq(ZXY.ma-3*y-5)\n几何问题的求解目标和状态（括号内数字表示目标状态，0表示此目标待求解，1表示此目标已求解，-1表示此目标不可能实现）：\nEq(y-15)(0)\n解析几何图形的结构信息得到的实体：\nPoint: (T), (X), (R), (Z), (S), (Y)\nLine: (Z,X), (R,S), (T,S), (R,T), (S,T), (Y,X), (Z,Y), (Y,Z), (X,Z), (S,R), (T,R), (X,Y)\nAngle: (X,Y,Z), (Z,X,Y), (Z,Y,X), (X,Z,Y), (S,R,T), (T,S,R), (Y,Z,X), (R,T,S), (S,T,R), (R,S,T), (Y,X,Z), (T,R,S)\nTriangle: (X,Y,Z), (Z,X,Y), (Y,Z,X), (S,T,R), (R,S,T), (T,R,S)\n按条件类型列出的所有已知条件：\nCongruentBetweenTriangle: (R,S,T,X,Y,Z), (S,T,R,Y,Z,X), (T,R,S,Z,X,Y)\n按照方程变量是否相交来分组，得到的代数方程组（所有方程省略'=0'、组序号可能不连续）：\nGroup 1: TR.ll-x-21, ZX.ll-2*x+14\nGroup 3: TRS.ma-4*y+10, ZXY.ma-3*y-5\n初始目标和所有分解得到的目标（括号内数字表示目标状态，0表示此目标待求解，1表示此目标已求解，-1表示此目标不可能实现）：\nEq(y-15)(0), 初始目标"
      },
      {
        "role": "assistant",
        "content": "{\n    \"thinking\": \"我们需要利用全等三角形的性质来得到对应边和对应角的相等关系。已知全等关系 CongruentBetweenTriangle(R,S,T,X,Y,Z) 以及其对称形式。为了得到 TR.ll = ZX.ll，我们可以使用 congruent_triangle_property_line_equal 定理，参数选择 (S,T,R,Y,Z,X)，因为此时 BC 边为 TR，EF 边为 ZX，正好对应。为了得到角 TRS.ma = ZXY.ma，可以使用 congruent_triangle_property_angle_equal 定理，参数选择 (R,S,T,X,Y,Z)，因为此时角 CAB 为 TRS，角 FDE 为 ZXY。先应用边相等的定理。\",\n    \"action\": \"apply(congruent_triangle_property_line_equal(S,T,R,Y,Z,X))\"\n}"
      },
      {
        "role": "user",
        "content": "工具执行结果：\n定理'congruent_triangle_property_line_equal(S,T,R,Y,Z,X)'执行成功，以下为问题的状态更新。\n新推导出的条件：\nEq(TR.ll-ZX.ll)\nEq(TR.ll-56)\nEq(ZX.ll-56)\nEq(x-35)"
      },
      {
        "role": "assistant",
        "content": "{\n    \"thinking\": \"使用 congruent_triangle_property_angle_equal 定理得到角 TRS 等于角 ZXY。\",\n    \"action\": \"apply(congruent_triangle_property_angle_equal(R,S,T,X,Y,Z))\"\n}"
      },
      {
        "role": "user",
        "content": "工具执行结果：\n定理'congruent_triangle_property_angle_equal(R,S,T,X,Y,Z)'执行成功，以下为问题的状态更新。\n新推导出的条件：\nEq(TRS.ma-ZXY.ma)\nEq(TRS.ma-50)\nEq(ZXY.ma-50)\nEq(y-15)\n部分目标的状态更新为（括号内数字表示目标状态，0表示此目标待求解，1表示此目标已求解，-1表示此目标不可能实现）：\nEq(y-15)(1)"
      },
      {
        "role": "user",
        "content": "检测到问题已求解，自动结束。"
      },
      {
        "role": "user",
        "content": "求解结束：成功✅"
      }
    ]
  ]
}

📄 许可证

MIT License

👤 作者

  • GitHub: @BitSecret
  • Email: xiaokaizhang@shu.edu.cn

🙏 致谢

感谢Datawhale社区、Hello-Agents项目以及FormalGeo项目！