*Ceva's* Theorem
was discovered by and named for Giovanni
Ceva. *Ceva's* Theorem states: Given any triangle ABC,
the segments from A, B, and C to the opposite sides of the triangle
are concurrent precisely when the product of the ratios of the
pairs of segments formed on each side of the triangle is equal
to 1. The theorem is easier to understand if you look at the following
picture (Figure 1). AB, BC, and CA are concurrent if AF/FB*BD/DC*CE/EA=1.
The converse is also true. If AF/FB*BD/DC*CE/EA=1 then AB, BC,
and CA are concurrent.

Figure 1

Because of *Ceva's* Theorem, the lines
joining a vertex of a triangle with a point on the opposite side
(AB, BC, and CA) are known as *Cevians
*(Bogmolny). Although Figure 1 only shows *Cevians* that
lie inside the triangle, *Ceva's *theorem is still
true when the *Cevians* do not all lie in the interior of
the triangle. While many people have
never heard of *Ceva's* theorem, most people have seen examples
of *Cevians* somewhere in geometry. For example, the medians of a triangle, the angle
bisectors of a triangle, and the altitudes
of a triangle are all examples of *Cevians*.

There are several proofs of *Ceva's* Theorem,
but the usual proof involves
extending segments of the triangle and considering the similar
triangles that are formed (Bogomolny).

A second proof of *Ceva's*
Theorem does not involve similar triangles. Instead, it uses the
theorem that the areas of triangles with equal altitudes are proportional
to the bases of the triangles (Coxeter, 4).

*Ceva's* Theorem
also gives rise to several Corollaries. The first three Corollaries
that arise from *Ceva's* Theorem have already been mentioned.
They are:

Corollary 1: The medians of a triangle intersect at a single point
(proof). The point of intersection
of the medians of a triangle is referred to as the centroid.

Corollary 2: In a triangle, the angle bisectors intersect at a
single point (proof). The intersection
point of the angle bisectors of a triangle is referred to as the
incenter of the triangle.

Corollary 3: In a triangle, the altitudes intersect at a single
point (proof).

The intersection point of the altitudes of a triangle is known
as the orthocenter of the triangle.

Another interesting Corollary to *Ceva's*
Theorem involves the *Gergonne* point. The Corollary states: Let D, E, and F be the points
where the inscribed circle touches the sides of the triangle ABC.
Then the lines AD, BE, and CF intersect at one point. *(essay 2: Gergonne Point)*

The last Corollary to *Ceva's* theorem
that we will mention involves the isotomic conjugates of concurrent
*Cevians*. The Corollary states: For three concurrent Cevians
AD, BE, and CF, if the point D, E, and F are reflected in the
midpoints of the corresponding sides, the resulting three lines
form another triplet of concurrent *Cevians*. The resulting
*Cevians *are referred to as isotomic
conjugates of the original *Cevians*. The proof of this
theorem is rather simple because the ratios are simply inverted.
So the product of the inverted ratios is still 1.

Within triangle ABC with *Cevians* AX,
BY, and CZ, there is the *Cevian* triangle and the *Cevian
*circle. The *Cevian *triangle is the triangle with vertices
at the endpoints of the *Cevians* through P (triangle XYZ).
The *Cevian* circle is the circumscribed circle of the *Cevian*
triangle. In other words it is the circle that passes through
all three vertices of the *Cevian* triangle. The *Cevian
*triangle and the *Cevian* circle are shown for triangle
ABC in figure 7 (Weisstein).

Figure 7

Besides *Ceva's* Theorem, Giovanni *Ceva*
is also credited with rediscovering and publishing *Menelaus'*
Theorem (JOC). *Menelaus'* Theorem is very closely related
to *Ceva's* Theorem. Both *Ceva's *Theorem and *Menelaus*'
Theorem use the statement BD/CD*CE/AE*AF/BF=1. *Ceva's* Theorem
states that if the three Cevians of a triangle are concurrent
then the previous statement holds. On the other hand, *Menelaus*'
Theorem states that if points D, E, and F on the sides BC, CA,
and AB of triangle ABC are collinear, then the previous statement
holds. Some references state *Menelaus'* Theorem in the following
way:** **given any line that transverses (crosses) the three
sides of a triangle (one of them will have to be extended), six
segments are cut off on the sides. The product of three non-adjacent
segments is equal to the product of the other three segments (Wilson).
Both statements of *Menelaus'* Theorem are equivalent. So
*Menelaus'* Theorem provides a criterion for collinearity,
while *Ceva's* Theorem provides a criterion for concurrence
(Coxeter, 67).

Like *Ceva's* Theorem, the proof
of *Menelaus' *Theorem also involves similar triangles
(No Author). Another proof of *Menelaus'*
Theorem involves using different similar triangles (Wilson).The
converse of *Menelaus'* Theorem is also true. If BD/CD*CE/AE*AF/BF=1
then D, E, and F are collinear.

A famous theorem that depends on *Menelaus'*
Theorem is *Desargues'* Theorem*. *Desargues'
Theorem states: Let ABC and A'B'C' be two triangles. If the
lines AA', BB', and CC' concur in one point, then the intersection
point of AB and A'B', the intersection point of BC and B'C', and
the intersection point of AC and A'C' are collinear. The converse
of this theorem is also true. If the three mentioned intersection
points are collinear, then AA', BB', and CC' concur in one point
(Floor).

Both *Ceva's *theorem and *Menelaus'*
theorem have been important contributions to the study of geometry.
*Ceva's *Theorem is used in the proofs of many well-known
theorems including the theorem of *Napolean's* point, the
theorem of *Fermat's* point, and the theorem of the *Nagel
*point. Because *Ceva's* Theorem and *Menelaus'*
Theorem deal with ideas that are so basic to the study of geometry,
concurrence and collinearity, it is impossible to overlook them.
However, many high school and college geometry courses do not
mention *Ceva's* Theorem or *Menelaus'* Theorem. But
none the less, the principles behind both of the theorems are
used and studied often in any geometry class.

Return to Essay 2: Gergonne Point