KNOWPIA
WELCOME TO KNOWPIA

In geometry, the **crossbar theorem** states that if ray AD is between ray AC and ray AB, then ray AD intersects line segment BC.^{[1]}

This result is one of the deeper results in axiomatic plane geometry.^{[2]} 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.^{[3]}

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.^{[4]}

**^**Greenberg 1974, p. 69**^**Kay 1993, p. 122**^**Blau 2003, p. 135**^**Moise 1974, p. 70

- Blau, Harvey I. (2003),
*Foundations of Plane Geometry*, Upper Saddle River, NJ: Prentice Hall, ISBN 0-13-047954-3 - Greenberg, Marvin J. (1974),
*Euclidean and Non-Euclidean Geometries*, San Francisco: W. H. Freeman, ISBN 0-7167-0454-4 - Kay, David C. (1993),
*College Geometry: A Discovery Approach*, New York: HarperCollins, ISBN 0-06-500006-4 - Moise, Edwin E. (1974),
*Elementary Geometry from an Advanced Standpoint*(2nd ed.), Reading, MA: Addison-Wesley, ISBN 0-201-04793-4