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]}