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 $\alpha :\left(1-\alpha \right)$ as $\alpha$ varies from 0 to $1/2$ . If $\alpha$ 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 $\alpha$ , 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.