The Crossbar Theorem says ray AD intersects segment BC
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]
In geometry, the crossbartheorem states that if ray AD is between ray AC and ray AB, then ray AD intersects line segment BC. This result is one of the...
the later tradition of Alexandria. In the Elements, Euclid deduced the theorems from a small set of axioms. He also wrote works on perspective, conic sections...
other subjects, coming up with the proof of what is now called Thales' Theorem. An equally enigmatic figure is Pythagoras of Samos (c. 580–500 BC), who...
proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( π {\displaystyle \pi } ) is a transcendental number...
be implemented by the individual crossbar switches in a Clos network with m = n. The proof uses Hall's marriage theorem which is given this name because...
Birkhoff's) and a continuity axiom which can be used to derive the Crossbartheorem and its converse. The increased number of axioms has the pedagogical...
bound on the log-evidence of the data. By the generalized Pythagorean theorem of Bregman divergence, of which KL-divergence is a special case, it can...
introduced in 1982 along with a neural network capable of self-learning, named crossbar adaptive array (CAA). It is learning with no external rewards and no external...
and Lempel–Ziv–Markov chain algorithm. Phillip Kuekes: inventor of the crossbar latch in semiconductor manufacturing. Dave Cliff: inventor of the ZIP trading...
direct distance dialing, E-repeater, wire spring relay, and the Number Five Crossbar Switching System. In 1952, William Gardner Pfann revealed the method of...
The VOQ mechanism provides throughput at a much higher rate than the crossbar switches without it. There are many algorithms for design and implementation...
24” printed circuit boards, with the network board using CSRD-designed crossbar gate array chips. The boards were assembled into three custom racks in...
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