Ceva's Theorem (pronounced "Cheva") is a very popular theorem in elementary geometry. Given a triangle ABC, and points D, E, and F that lie on lines BC, CA, and AB respectively, the theorem states that lines AD, BE and CF are concurrent if and only if Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ... Parallel programming (also concurrent programming), is a computer programming technique that provides for the execution of operations concurrently, either within a single computer, or across a number of systems. ... In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...

It was first proven by Giovanni Ceva. Giovanni Ceva (1648-1734) is an Italian mathematician widely known for proving Cevas Theorem in elementary geometry. ...

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Proof

Suppose AD, BE and CF intersect at a point X. Because and have the same height, we have

Similarly,

From this it follows that

Similarly,

Multiplying these three equations gives

as required. Conversely, suppose that the points D, E and F satisfy the above equality. Let AD and BE intersect at X, and let CX intersect AB at F'. By the direction we have just proven,

Comparing with the above equality, we obtain

Adding 1 to both sides and using AF' + F'B = AF + FB = AB, we obtain

Thus F'B = FB, so that F and F' coincide (recalling that the distances are directed). Therefore AD, BE and CF=CF' intersect at X, and both implications are proven.

See also

• Menelaus theorem

Menelaus theorem (also known as Menelaus theorem, Menelauss theorem, as well as theorem of Menelaus; attributed to Menelaus of Alexandria) is a theorem about triangles in plane geometry. ...

External links

• Ceva's Theorem, Interactive proof with animation and key concepts by Antonio Gutierrez from the land of the Incas
• Derivations and applications of Ceva's Theorem
• Cevian Nest
• Trigonometric Form of Ceva's Theorem