**A Proof of Ceva’s Theorem**

**By**

## Ken Montgomery

,
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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