The Theorems of Menelaus and Ceva

The Theorem of Menelaus and Ceva's Theorem are very closely related. Both concern the products of ratios of lengths involving lines cutting off parts of a triangle. However, the theorem of Menelaus is about 1600 years older than Ceva's theorem. Menelaus of Alexandria was born about 70 AD, while Giovanni Ceva lived between 1647 and 1734.

In our discussion here, we will only briefly state the theorems. For more details and proofs, see the very nice discussion at Cut The Knot and/or your textbook. This page requires a java-enabled browser for correct functioning. You can drag the points labelled A, B, C, P, and Q around with the mouse, and the rest of the picture will change accordingly.

Theorem of Menelaus

Let three points X, Y, and Z, lie respectively on the sides AC, BC, and AB of triangle ABC. Then the points are collinear if and only if

AZ/ZB * CX/XA * BY/YC = -1

Note that these distances are signed, so if Z lies beyond B, the ratio AZ/ZB will be negative because ZB goes in the opposite direction from AZ.

In the applet at right, it wasn't possible to calculate signed distances, so the product is positive.

Please enable Java for an interactive construction (with Cinderella).
Ceva's Theorem

Three Cevians AY, BX, and CZ are concurrent at a point P if and only if

|AZ|/|ZB| * |CX|/|XA| * |BY|/|YC| = 1

A cevian is a segment from a vertex of a triangle to any point on the side opposite except another vertex. Note that this point need not be interior to the triangle.

For Ceva's Theorem, we can use either signed distances or not; the result is the same since there will always be an even number of negative distances (zero if P is inside the triangle, two if it is outside).

Please enable Java for an interactive construction (with Cinderella).

Java images created using Cinderella by Scott Sutherland on March 15, 2004.