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In 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 c ...
, the crossbar theorem states that if ray AD is between ray
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (gra ...
AC and ray AB, then ray AD intersects 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 ...
BC.
This result is one of the deeper results in axiomatic plane geometry. It is often used in proofs to justify the statement that a line through a vertex of a triangle lying ''inside'' the triangle meets the side of the triangle opposite that vertex. This property was often used by Euclid in his proofs without explicit justification.
Some modern treatments (not Euclid's) of the proof of the theorem that the base angles of an isosceles triangle are congruent start like this: Let ABC be a triangle with side AB congruent to side AC. ''Draw the angle bisector of angle A and let D be the point at which it meets side BC''. And so on. The justification for the existence of point D is the often unstated crossbar theorem. For this particular result, other proofs exist which do not require the use of the crossbar theorem.
See also
*Foundations of geometry
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but ...
* Jordan curve theorem
In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exterio ...
Notes
References
* * * * Euclidean plane geometry Theorems in plane geometry Foundations of geometry {{Elementary-geometry-stub