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.
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