geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a cleaver of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...

is a

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...

that bisects the

perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pract ...

of the triangle and has one endpoint at the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimen ...

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 Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists i ...

provides another construction of the cleaver. Suppose the triangle to be bisected is , and that one endpoint of the cleaver is the midpoint of side . Form the circumcircle of and let be the midpoint of the arc of the circumcircle from to through . Then the other endpoint of the cleaver is the closest point of the triangle to , and can be found by dropping a perpendicular from to the longer of the two sides and .

Related figures

The three cleavers

concur In Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liabilit ...

at a point, the the center of the

Spieker circle In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the cent ...

References

External links

*{{mathworld, title=Cleaver, urlname=Cleaver, mode=cs1 Straight lines defined for a triangle

Construction

Related figures

See also

References

External links