The existence of a triangle when its three elements are known

Abstract

The problem of the existence of a triangle with respect to three given elements in some cases can be very difficult. For example, Brokard's problem about the existence of a triangle, given its three bisectors [1], has a long history [3] and solved only in 1994 [10]. We include in the number of elements: three sides, three angles, three heights, three medians, three bisectors, radii of the circumscribed and inscribed circles, and perimeter. In total, there are 186 different problems of the existence of a triangle with three given elements and for 116 problems are given sufficient conditions (for some sufficient and necessary conditions of existence) when a triangle can be construct by a compass and a ruler, and the remaining 70 problems when it is impossible to construct a triangle by a compass and a ruler. The authors list these 70 problems and indicate for which of them the necessary and sufficient conditions for the uniqueness of the existence of a triangle with three prescribed elements have found.

Keywords

• triangle
• solution of triangles
• lexicographic order
• height; median
• bisector
• radii of the circumscribed and inscribed circles