Dividing the Perimeter of a Triangle into Unequal Proportions

Abstract

We fully describe the envelope of all line segments that divide the perimeter of a triangle into the ratio α:1-α as α varies from 0 to 1/2. If α is larger than the ratio of the longest side length to the perimeter, then the envelope is a 12-sided closed curve consisting of six line segments and six parabolic arcs. For other values of α, the envelope is the union of one to three parabolic arcs and possibly a 5- or 9-sided nonclosed curve consisting of line segments and parabolic arcs. © 2022 Nawinda Amadtohed et al.