Ceva's Theorem

Ceva's Theorem is a criterion for the concurrence of cevians in a triangle.

Statement

Let $ABC$ be a triangle, and let $D, E, F$ be points on lines $BC, CA, AB$ , respectively. Lines $AD, BE, CF$ concur iff (if and only if)

$\frac{BD}{DC} \cdot \frac{CE}{EA}\cdot \frac{AF}{FB} = 1$ ,

where lengths are directed. This also works for the reciprocal or each of the ratios, as the reciprocal of $1$ is $1$ .

(Note that the cevians do not necessarily lie within the triangle, although they do in this diagram.)

Proof

We will use the notation $[ABC]$ to denote the area of a triangle with vertices $A,B,C$ .

First, suppose $AD, BE, CF$ meet at a point $X$ . We note that triangles $ABD, ADC$ have the same altitude to line $BC$ , but bases $BD$ and $DC$ . It follows that $\frac {BD}{DC} = \frac{[ABD]}{[ADC]}$ . The same is true for triangles $XBD, XDC$ , so

$\frac{BD}{DC} = \frac{[ABD]}{[ADC]} = \frac{[XBD]}{[XDC]} = \frac{[ABD]- [XBD]}{[ADC]-[XDC]} = \frac{[ABX]}{[AXC]}$ .

Similarly, $\frac{CE}{EA} = \frac{[BCX]}{[BXA]}$ and $\frac{AF}{FB} = \frac{[CAX]}{[CXB]}$ , so

$\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = \frac{[ABX]}{[AXC]} \cdot \frac{[BCX]}{[BXA]} \cdot \frac{[CAX]}{[CXB]} = 1$ .

Now, suppose $D, E,F$ satisfy Ceva's criterion, and suppose $AD, BE$ intersect at $X$ . Suppose the line $CX$ intersects line $AB$ at $F'$ . We have proven that $F'$ must satisfy Ceva's criterion. This means that

$\frac{AF'}{F'B} = \frac{AF}{FB}$ ,

so

$F' = F$ ,

and line $CF$ concurrs with $AD$ and $BE$ . ∎

Trigonometric Form

The trigonometric form of Ceva's Theorem (Trig Ceva) states that cevians $AD,BE,CF$ concur if and only if

$\frac{\sin BAD}{\sin DAC} \cdot \frac{\sin CBE}{\sin EBA} \cdot \frac{\sin ACF}{\sin FCB} = 1.$

Proof

First, suppose $AD, BE, CF$ concur at a point $X$ . We note that

$\frac{[BAX]}{[XAC]} = \frac{ \frac{1}{2}AB \cdot AX \cdot \sin BAX}{ \frac{1}{2}AX \cdot AC \cdot \sin XAC} = \frac{AB \cdot \sin BAD}{AC \cdot \sin DAC}$ ,

and similarly,

$\frac{[CBX]}{[XBA]} = \frac{BC \cdot \sin CBE}{BA \cdot \sin EBA} ;\; \frac{[ACX]}{[XCB]} = \frac{CA \cdot \sin ACF}{CB \cdot \sin FCB}$ .

It follows that

$\frac{\sin BAD}{\sin DAC} \cdot \frac{\sin CBE}{\sin EBA} \cdot \frac{\sin ACF}{\sin FCB} = \frac{AB \cdot \sin BAD}{AC \cdot \sin DAC} \cdot \frac{BC \cdot \sin CBE}{BA \cdot \sin EBA} \cdot \frac{CA \cdot \sin ACF}{CB \cdot \sin FCB}$

$\qquad = \frac{[BAX]}{[XAC]} \cdot \frac{[CBX]}{[XBA]} \cdot \frac{[ACX]}{[XCB]} = 1$ .

Here, sign is irrelevant, as we may interpret the sines of directed angles mod $\pi$ to be either positive or negative.

The converse follows by an argument almost identical to that used for the first form of Ceva's Theorem. ∎

Problems

Introductory

Suppose $AB, AC$ , and $BC$ have lengths $13, 14$ , and $15$ , respectively. If $\frac{AF}{FB} = \frac{2}{5}$ and $\frac{CE}{EA} = \frac{5}{8}$ , find $BD$ and $DC$ . (Ceva's Theorem/Problems)

Intermediate

Olympiad

Other Notes

The concurrence of the altitudes of a triangle at the orthocenter and the concurrence of the perpendicual bisectors of a triangle at the circumcenter can both be proven by Ceva's Theorem (the latter is a little harder). Furthermore, the existance of the centroid can be shown by Ceva, and the existance of the incenter can be shown using trig Ceva. However, there are more elegant methods for proving each of these results, and in any case, any result obtained by classic Ceva's Theorem can be proven using ratios of areas.
The existance of isotonic conjugates can be shown by classic Ceva, and the existance of isogonal conjugates can be shown by trig Ceva.

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Ceva's Theorem

Contents

Statement

Proof

Trigonometric Form

Proof

Problems

Introductory

Intermediate

Olympiad

Other Notes

See also