On-line compendiums of triangle location construction problems
Triangle location construction problems are construction problems in which the task is to construct a triangle ABC, using straightedge and compass, given positions of three characteristic points of the triangle. In 1982. William Wernick presented a list of triangle location problems where three characteristic points are selected among the following 16 points:
- A, B, C, O: three vertices and circumcenter;
- Ma, Mb, Mc, G: the side midpoints and centroid;
- Ha, Hb, Hc, H: three feet of altitudes and orthocenter;
- Ta, Tb, Tc, I: three feet of the internal angle bisectors and incenter.
Wernick's list consists of 139 significantly different problems, and today statuses of all problems from Wernick's list are known: there are 74 solvable problems, 39 unsolvable problems, 3 redundant problems and 23 locus dependent problems.
Harold Connelly considered an extended Wernick's list, involving four additional points:
- Ea, Eb, Ec: three Euler points;
- N: the center of nine point circle.
Current status of problems from Connelly's list is the following: there are 73 solvable problems, 11 unsolvable problems, 5 redundant problems, 19 locus dependent problems and 32 problems with unknown status.
Generated compendiums
Compendium of solutions to problems from extended Wernick's corpus
Compendium of solutions to problems from extended Connelly's corpus
Bibliography
- William Wernick, Triangle Constructions with Three Located Points, Mathematics Magazine 55, no. 4, 1982.
- Harold Connelly, An Extension of Triangle Constructions from Located Points, Forum Geometricorum 9, 2009.
- Vesna Marinkovic, Predrag Janicic, Towards Understanding Triangle Construction Problems, Intelligent Computer Mathematics - CICM 2012. Lecture Notes in Computer Science, vol. 7362, Springer, 2012.
- Vesna Marinkovic, Pascal Schreck, Predrag Janicic, Computer Theorem Proving for Verifiable Solving of Geometric Construction Problems, Automated Deduction in Geometry 2014, LNCS 9201, Springer, 2015.
- Vesna Marinkovic, ArgoTriCS - Automated Triangle Construction Solver, submitted.
System for automated solving construction problems ArgoTriCS
Vesna Marinkovic, The Faculty of Mathematics, University of Belgrade