Cleaver (geometry)

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.

[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.

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

Construction of the Spieker center via cleavers.
Triangle △ ABC
Angle bisectors of △ ABC ( concurrent at the incenter I )
Cleavers of △ ABC (concurrent at the Spieker center S )
Medial triangle △ DEF of △ ABC
Inscribed circle of △ DEF (the Spieker circle ; centered at S )