In geometry, a **cleaver** of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides.

## Construction

Each cleaver through the midpoint of one of the sides of a triangle is parallel to the angle bisectors at the opposite vertex of the triangle.^{[1]}^{[2]}

The **broken chord theorem** of Archimedes provides another construction of the cleaver. Suppose the triangle to be bisected is *ABC*, and that one endpoint of the cleaver is the midpoint of side *AB*. Form the circumcircle of *ABC* and let *M* be the midpoint of the arc of the circumcircle from *A* to *B* through *C*. Then the other endpoint of the cleaver is the closest point of the triangle to *M*, and can be found by dropping a perpendicular from *M* to the longer of the two sides *AC* and *BC*.^{[1]}^{[2]}

## Related figures

The three cleavers concur at a point, the center of the Spieker circle.^{[1]}^{[2]}

## See also

## References

- ^
^{a}^{b}^{c}Honsberger, Ross (1995), "Chapter 1: Cleavers and Splitters",*Episodes in Nineteenth and Twentieth Century Euclidean Geometry*, New Mathematical Library,**37**, Washington, DC: Mathematical Association of America, pp. 1–14, ISBN 0-88385-639-5, MR 1316889 - ^
^{a}^{b}^{c}Avishalom, Dov (1963), "The perimetric bisection of triangles",*Mathematics Magazine*,**36**(1): 60–62, JSTOR 2688140, MR 1571272