Menelaus's Theorem is very similar to Ceva's
Theorem. The theorem states that if a straight line intersects
all three sides of a triangle (one or all three intersections may be on
the extended legs) then the sides
must be cut into proportions that multiply to make one. Using the
figure, triangle ABC is cut by the line at A', B', and C' on the opposite
sides of the trinangle and so .
The theorem is also written in the equivalent form, .
The theorem is named for Menelaus of Alexandria,
who lived around the end of the first century. You can find more
about his life at the St
Andrews University web site.
Menelaus also proved a spherical version
of the same theorem. If the triangle and the cutting line are all
formed by great circle arcs on a sphere, then the formula states that .
Almost 1900 years after Menelaus (1995) a mathematician
named Hoehn published a similar looking theorem about Pentagons in Mathematics
Magazine. In it he stated that the product of all the red segments
equals the product of all the blue segments.