Ceva's theorem

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

It was first proved by Giovanni Ceva.

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

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