Triangle Equalizers

@article{Kodokostas2010TriangleE,
  title={Triangle Equalizers},
  author={D. Kodokostas},
  journal={Mathematics Magazine},
  year={2010},
  volume={83},
  pages={141 - 146}
}
Summary A triangle equalizer is a line bisecting both its area and perimeter. We provide a detailed account of equalizer locations, showing that there exist triangles with exactly one, two, or three equalizers, but no more. Triangles with exactly two equalizers are quite rare: Their smallest angle is less or equal to a particular angle of approximately 49 degrees, and the angle next in size is the unique root of a trigonometric equation in a specific interval depending on the smallest angle. 
1 Citations
Lines Simultaneously Bisecting the Perimeter and Area of a Triangle
Given triangle ABC with sidelengths BC = a, CA = b, and AB = c, let s denote the semiperimeter. Let Y and Z be points on the sides AC and AB respectively. We call the line Y Z (i) a perimeterExpand

References

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Episodes in Nineteenth and Twentieth Century Euclidean Geometry
Preface Introduction 1. Cleavers and splitters 2. The orthocenter 3. On triangles 4. On quadrilaterals 5. A property of triangles 6. The Fuhrmann circle 7. The symmedian point 8. The Miquel theoremExpand
The equalizer of a triangle, a clever line that does double duty, Quantum 7 (March–April
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The equalizer of a triangle, a clever line that does double duty