An Extension of Triangle Constructions from Located Points

Abstract. W. Wernick has tabulated 139 triangle construction problems using a list of sixteen points associated with the triangle. We add four points to his list and find an additional 140 construction problems. William Wernick [3] and Leroy Meyers [2] discussed the problem of construct-ing a triangle with ruler and compass given the location of three points associated with the triangle. Wernick tabulated all the significantly distinct problems that could be formed from the following list of sixteen points: A, B, C Three vertices Ma, Mb, Mc Three midpoints of the sides Ha, Hb, Hc Three feet of the altitudes Ta, Tb, Tc Three feet of the internal angle bisectors G, H, I, O The centroid, orthocenter, incenter and circumcenter Wernick found 139 triples that could be made from these points. They can be divided into the following four distinct types: R – Redundant. Given the location of two of the points of the triple, the location of the third point is determined. An example would be: A, B, Mc. L – Locus Restricted. Given the location of two points, the third must lie on a certain locus. Example: A, B, O. S – Solvable. Known ruler and compass solutions exist for these triples. U – Unsolvable. By using algebraic means, it is possible to prove that no ruler and compass solution exists for these triples. Example: O, H, I; see [1, 4]. To extend the work of Wernick and Meyers, we add the following four points to their list: Ea, Eb, Ec Three Euler points, which are the midpoints between the vertices and the orthocenter N The center of the nine-point circle. Tabulated below, along with their types, are all of the 140 significantly distinct triples that can be formed by adding our new points to the original sixteen. Prob-lems that remain unresolved as to type are left blank. In keeping with the spiri