Cleaver (geometry)

File:CleavanceCenter.svg

Construction of the Spieker center via cleavers.

  Triangle △ABC

  Angle bisectors of △ABC (concurrent at the incenter Template:Mvar)

  Cleavers of △ABC (concurrent at the Spieker center Template:Mvar)

  Medial triangle △DEF of △ABC

  Inscribed circle of △DEF (the Spieker circle; centered at Template:Mvar)

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. They are not to be confused with splitters, which also bisect the perimeter, but with an endpoint on one of the triangle's vertices instead of its 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.

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 Template:Mvar. Form the circumcircle of △ABC and let Template:Mvar be the midpoint of the arc of the circumcircle from Template:Mvar through Template:Mvar to Template:Mvar. Then the other endpoint of the cleaver is the closest point of the triangle to Template:Mvar, and can be found by dropping a perpendicular from Template:Mvar to the longer of the two sides Template:Mvar and Template:Mvar.

Related figures

The three cleavers concur at a point, the center of the Spieker circle.

See also

• Splitter (geometry)