We present a method for automatically building diagrams for olympiad-level geometry problems and implement our approach in a new open-source software tool, the Geometry Model Builder (GMB). Central to our method is a new domain-specific language, the Geometry Model-Building Language (GMBL), for specifying geometry problems along with additional metadata useful for building diagrams. A GMBL program specifies (1) how to parameterize geometric objects (or sets of geometric objects) and initialize these parameterized quantities, (2) which quantities to compute directly from other quantities, and (3) additional constraints to accumulate into a (differentiable) loss function. A GMBL program induces a (usually) tractable numerical optimization problem whose solutions correspond to diagrams of the original problem statement, and that we can solve reliably using gradient descent. Of the 39 geometry problems since 2000 appearing in the International Mathematical Olympiad, 36 can be expressed in our logic and our system can produce diagrams for 94% of them on average. To the best of our knowledge, our method is the first in automated geometry diagram construction to generate models for such complex problems.

中文翻译：

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自动构建奥林匹克几何问题的图表

我们提出了一种用于自动构建奥林匹克级别的几何问题的图的方法，并在新的开源软件工具Geometry Model Builder（GMB）中实现了我们的方法。我们方法的核心是一种新的领域特定语言，即几何模型构建语言（GMBL），用于指定几何问题以及对构建图表有用的其他元数据。GMBL程序指定（1）如何参数化几何对象（或一组几何对象）并初始化这些参数化的量，（2）要直接从其他量计算出的量，以及（3）附加约束累积到（可微分）损失函数。GMBL程序会引发一个（通常）易于处理的数值优化问题，其解对应于原始问题陈述的图，并且我们可以使用梯度下降可靠地解决。自2000年以来，在国际数学奥林匹克中出现的39个几何问题中，有36个可以用我们的逻辑表示，我们的系统平均可以为94％的问题生成图表。据我们所知，我们的方法是自动几何图构造中第一个为此类复杂问题生成模型的方法。