Solving Geometry Problems Using a Combination of Symbolic and Numerical Reasoning

  • Shachar Itzhaky
  • Sumit Gulwani
  • Neil Immerman
  • Mooly Sagiv
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8312)


We describe a framework that combines deductive, numeric, and inductive reasoning to solve geometric problems. Applications include the generation of geometric models and animations, as well as problem solving in the context of intelligent tutoring systems.

Our novel methodology uses (i) deductive reasoning to generate a partial program from logical constraints, (ii) numerical methods to evaluate the partial program, thus creating geometric models which are solutions to the original problem, and (iii) inductive synthesis to read off new constraints that are then applied to one more round of deductive reasoning leading to the desired deterministic program. By the combination of methods we were able to solve problems that each of the methods was not able to solve by itself.

The number of nondeterministic choices in a partial program provides a measure of how close a problem is to being solved and can thus be used in the educational context for grading and providing hints.

We have successfully evaluated our methodology on 18 Scholastic Aptitude Test geometry problems, and 11 ruler/compass-based geometry construction problems. Our tool solved these problems using an average of a few seconds per problem.


geometry reasoning synthesis 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Shachar Itzhaky
    • 1
  • Sumit Gulwani
    • 2
  • Neil Immerman
    • 3
  • Mooly Sagiv
    • 1
  1. 1.Tel Aviv UniversityIsrael
  2. 2.Microsoft ResearchRedmondUSA
  3. 3.University of MassachusettsAmherstUSA

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