Dividing the perimeter of a triangle into unequal proportions (Q2081849)

Summary: 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.