Menelaus' theorem

The set of Menelaus, named after the Greek mathematician Menelaus (Alexandria, about 100 AD), makes a statement about straight lines intersecting triangles.

Given a triangle ABC and a straight line which intersects the sides of the triangle [BC ] [CA ] and [ O], or their extensions at points X, Y and Z. Then:

Conversely, one can infer from the truth of this relationship that the points X, Y and Z lie on a straight line.

Here, the division ratio of which is defined on three located on a straight line through points. If is between and, this part ratio is equal, otherwise the same.

If only the path lengths, so you can write the above equation in the following form:

Since the orientation is lost in this case, this equation is not sufficient for a reversal of the sentence, see Theorem of Ceva.

Evidence

The set of Menelaus can be proved using the intercept theorem. We considered three solders on the given line, emanating from the vertices A, B and C. The lengths of the solder drawing are denoted by a, b and c.

From the theorem we obtain the following relation equations:

Multiplying these three equations together, we obtain

And further ( by multiplying by the denominator)

Application

Together with its inverse, the set of Menelaus provides a criterion for collinear points. One implication is the set of Ceva.