A Proof of Ceva’s Theorem

By

Ken Montgomery

Ceva’s Theorem:

, and intersect in one point, T if and only if:

Proof:

Case 1: T lies inside 

Let D lie on  , E lie on , F on  and let , and intersect at T (Figure 1).

Figure 1:  , with T in the triangle’s interior

Then, we have that:

However,

and since

we obtain

Likewise, for side , we have:

but since,

and since,

we have

Furthermore, for side we also have:

but since,

and since,

we have

Therefore, we have

Conversely, assume that

.

For each point, M on , and only for points on , we have that

Further, the line consists of points N, satisfying the equation:

Also, consists of points O, satisfying the equation:

Let T be the point of intersection of and . Thus,

and we have

so T lies on also.

Case 2: Without loss of generality, let T lie on the opposite side of from C.

Let D lie outside of  , E lie outside of , F on  and let , and intersect at T (Figure 2).

Figure 2:  , with T opposite of  , from C

Then, we have, for side that:

However, since

and since

we have

Likewise, for side we have that:

However, since

and since

we have

Also, for side , we have that:

However, since

and since

we have

Therefore, we have the equation

Conversely, assume that

then it is also true that

For each point, M on , and only for points on , we have that

Further, the line consists of points N, satisfying the equation:

Also, consists of points O, satisfying the equation:

Let T be the point of intersection of and . Thus,

and we have

so T lies on also.

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