Ceva's Theorem

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

Statement

Let be a triangle, and let be points on lines , respectively. Lines are concurrent if and only if

,

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

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

The proof using Routh's Theorem is extremely trivial, so we will not include it.

Proof

We will use the notation to denote the area of a triangle with vertices .

First, suppose meet at a point . We note that triangles have the same altitude to line , but bases and . It follows that . The same is true for triangles , so

.

Similarly, and , so

.

Now, suppose satisfy Ceva's criterion, and suppose intersect at . Suppose the line intersects line at . We have proven that must satisfy Ceva's criterion. This means that

,

so

,

and line concurrs with and . ∎

Proof by Barycentric coordinates

Since , we can write its coordinates as . The equation of line is then .

Similarly, since , and , we can see that the equations of and respectively are and

Multiplying the three together yields the solution to the equation:

Dividing by yields:

, which is equivalent to Ceva's theorem

QED

Trigonometric Form

The trigonometric form of Ceva's Theorem (Trig Ceva) states that cevians concur if and only if

Proof

First, suppose concur at a point . We note that

,

and similarly,

.

It follows that

.

Here, sign is irrelevant, as we may interpret the sines of directed angles mod 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 , and have lengths , and , respectively. If and , find and . (Source)

Intermediate

• In are concurrent lines. are points on such that are concurrent. Prove that (using plane geometry) are concurrent. (<url>viewtopic.php?f=151&t=543574 Source + Answer<url>)