![](bg2.png)
etry questions based on specific inputs, such as geom-
etry objects, theorems and construction objects to be
involved in their solution. For a given set of geome-
try objects, the algorithm can generate a large set of
questions along with their solutions. The solutions
will involve user desired theorems directly or indi-
rectly. Hence the framework can generate questions
to test the theorem on various geometry objects and
concepts.
Generated questions from our framework having
implicit construction may involve algebraic computa-
tions for a solution. Currently, we restrict the rela-
tionships between quantitative entities to linear ones.
Our framework has a predefined database of theo-
rems, concepts and construction objects which can be
used for generating questions. Given a set of user-
selected construction objects, our system can gener-
ate all possible questions using an existing database
of concepts and theorems.
The main contributions of this paper are as fol-
lows:
1. Our geometry question generator combines the
complementary strengths of a combinatorial ap-
proach, pattern matching and deductive reason-
ing. Construction-based geometry questions can
be generated which were not possible previously.
2. A substantial evaluation is provided that demon-
strates the effectiveness of our generator. The
question generator was able to generate most
of the questions, involving implicit construc-
tion, covered in the textbooks based on targeted
queries.
2 RELATED WORK
There are several automated theorem proving system
in the geometric domain such as JGEX (Gao and Lin,
2004), Cinderella and Geometry Explorer (Wilson
and Fleuriot, 2006) that allows users to build proofs
based on geometric constructions. Angle method
(Gao and Lin, 2004), Wu’s method and Grobner basis
method (Chou et al., 1994) are some of the methods
used by the existing geometry systems for automat-
ically proving geometric theorems. However, these
methods are either not suitable for question genera-
tion or the approach used is not suitable for the high
school geometric domain. Hence, the geometric sys-
tems are unable to automatically generate geometric
questions.
GRAMY (Matsuda and Vanlehn, 2004) is a ge-
ometric system that can solve geometric questions
which require implicit construction. It uses a for-
ward chaining search followed by a backward step to
suggest a construction. GRAMY can only discover
proofs that do not involve arithmetic operations. For
example, it cannot find proofs that involve inequali-
ties and ratios.
An algorithm developed by Rohit (R. Singh and
Rajamani, 2012) used a numerical approach for solv-
ing geometric questions involving construction. The
algorithm uses the concept of randomness instead of
performing symbolic reasoning. Hence, the solutions
generated by this algorithm are based on numerical
reasoning and out of the scope of high school mathe-
matics.
Our original framework can generate geometric
questions based on concepts, theorems and user-
selected geometric objects. However, it cannot gen-
erate construction based questions. Current work is
an extension of the framework developed in (annony-
mous, 2013).
3 GEOMETRY QUESTION
GENERATION TASK
Mathematically a geometry question Q generated by
our system can be represented by a sextuple (Ob-
ject O, Concept C, Theorem T, Construction objects
CObj, Relationship R, Query type Qt) where:
• O ∈ (lines, triangles, square, circle, ... )
• C ∈ (perpendicular, parallel, midpoint, angle-
bisector, circumcircle...)
• T ∈ (Pythagorean theorem, similarity theorem,
various triangle-theorems, ...)
• CObj ∈ (perpendicular, parallel, midpoint, angle-
bisector...)
• R ∈ (syntactic, quantitative)
• Qt ∈ (syntactic, quantitative)
In order to generate geometry questions, the user has
to provide a set of geometry objects O such as tri-
angles, squares, etc., a set of concepts C and a set
of construction objects CObj such as perpendicular,
parallel, midpoint, etc., which the user would like to
cover in the generated question. Optionally the user
may select a set of theorems T to be tested by the
question. The relationship R can be either syntac-
tic such as perpendicular, parallel, etc., or quantita-
tive such as the giving the length of an object, the ra-
tio of two quantities etc. The query type Qt is the
type of generated question that can be asked to find
the hidden relationship which can be calculated from
the given information. The solution of the generated
question would require construction of user-selected
CSEDU2014-6thInternationalConferenceonComputerSupportedEducation
468