Ceva's theorem 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

$\backslash frac\{AF\}\{FB\}\; \backslash cdot\; \backslash frac\{BD\}\{DC\}\; \backslash cdot\; \backslash frac\{CE\}\{EA\}\; =\; 1.$

There is also an equivalent trigonometric form of Ceva's Theorem, that is, AD,BE,CF concur if and only if
$\backslash frac\{\backslash sin\backslash angle\; BAD\}\{\backslash sin\backslash angle\; CAD\}\backslash times\backslash frac\{\backslash sin\backslash angle\; ACF\}\{\backslash sin\backslash angle\; BCF\}\backslash times\backslash frac\{\backslash sin\backslash angle\; CBE\}\{\backslash sin\backslash angle\; ABE\}=1$.

It was first proven by Giovanni Ceva in his 1678 work De lineis rectis.

Proof

Suppose $AD$, $BE$ and $CF$ intersect at a point $O$. Because $\backslash triangle\; BOD$ and $\backslash triangle\; COD$ 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 $O$, and let $CO$ intersect $AB$ at $F\text{'}$. By the direction we have just proven,

Comparing with the above equality, we obtain

Adding 1 to both sides and using $AF\text{'}+F\text{'}B=AF+FB=AB$, we obtain

Thus $F\text{'}B=FB$, so that $F$ and $F\text{'}$ coincide (recalling that the distances are directed). Therefore $AD$, $BE$ and $CF$=$CF\text{'}$ intersect at $O$, 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 at cut-the-knot
• Cevian Nest at cut-the-knot
• Trigonometric Form of Ceva's Theorem at cut-the-knot

Affine geometry | Euclidean plane geometry | Mathematical theorems

Satz von Ceva | Théorème de Ceva | 체바의 정리 | Teorema di Ceva | チェバの定理 | Twierdzenie Cevy | Теорема Чевы | Cevov izrek | Cevan lause | 塞瓦定理