Ivan Petrović

Computer Science DepartmentFaculty of MathematicsUniversity of Belgrade

February 5th, 2011

Java implementation of Wu's method for

Automated Theorem Proving in Geometry

Geometry Theorem Provers 1/11

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Two categories of provers: algebraic (coordinate-based) methods coordinate-free methods

Main algebraic methods: Wu's method (Wen-Tsun Wu) Gröbner bases method (Bruno

Buchberger)

Main coordinate-free methods: Area method (Shang-Ching Chou, Xiao-

Shan Gao, Jing-Zhong Zhang) Full-Angle method (same authors)

Geometry Theorem Provers 2/11

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Wu's method is powerful mechanism for proving geometry theorems in elementary geometry. It is complete decision procedure for some classes of geometry problems.

How Wu's method works? step 1 – translate geometry problem

into multivariate polynomial system two types of variables: us – independent (parametric) variables xs – dependent variables step 2 – triangulation of polynomial

system (each next equation introduces exactly one new dependent variable) by using pseudo division

Geometry Theorem Provers 3/11

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step 3 – calculating final reminder of polynomial that represents statement with each polynomial from triangulated system, by using pseudo division of polynomials

step 4 – producing answer on the basis of final reminder and obtained non-degenerative conditions (zero reminder means proved theorem)

Geometry Theorem Provers 4/11

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main operation – pseudo division:

Geometry Theorem Provers 5/11

_________________________Wu's method in WinGCLC application

(screen shot of Euler's line theorem)

Geometry Theorem Provers 6/11

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Simple example of Wu's method:

[Theorem about circumcenter of a triangle]

“The tree perpendicular bisectors of a triangle's sides meet in a single point (they are concurrent lines).”

Geometry Theorem Provers 7/11

_________________________Construction written in GCLC:

point A 20 20cmark_b Apoint B 50 20cmark_b B point C 40 70cmark_t C

drawsegment A Bdrawsegment B Cdrawsegment C A

med mab A Bmed mac A Cmed mbc B C

drawline mabdrawline macdrawline mbc

intersec M_1 mab macintersec M_2 mab mbc

cmark_rt M_1cmark_lb M_2

prove {identical M_1 M_2}

Geometry Theorem Provers 8/11

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Prover output clipping for this example

Geometry Theorem Provers 9/11

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Prover result for this example

Geometry Theorem Provers 10/11

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Reimplementation in Java programming language(based on C++ version by Goran Predović and Predrag

Janičić)

Main objectives of this project: greater portability ability of integration in other systems for

mechanical theorem proving and geometry related software (GeoGebra, Geo Thms etc)

Directions for further work: possible improvements of current

implementation by usage of concurrency implementing Gröbner bases prover

Geometry Theorem Provers 11/11

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Current state of this project:

Classes for algebraic primitives are almost completed

Prepared utilities for prover output to LaTeX and XML format

Implemented pseudo reminder algorithm; after implementation of simple triangulation algorithm, Wu's method is almost completed

At the end dealing with transformation of GCLC input into polynomial form

Geometry Theorem Provers The End

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Thank you.